INTERNATIONAL GONGRESS . The main point is that the notion of a finite correspondence for smooth finite type schemes over a field extends to a corresponding notion over a general base-scheme (see [33,$8][33,$8][33,$8][33, \$ 8][33,$8] ). This gives rise to a theory of motivic cohomology generalizing Voevodsky's definition as
for XXXXX smooth over SSSSS. They show that the assignment S↦DMCD(S)S↦DMCDâ¡(S)S|->DM_(CD)(S)S \mapsto \operatorname{DM}_{\mathrm{CD}}(S)S↦DMCDâ¡(S) defines a functor to the category of triangulated tensor categories, DMCD(−):SchBop→Tr⊗DMCD(−):SchBop→Tr⊗DM_(CD)(-):Sch_(B)^(op)rarrTr^(ox)\mathrm{DM}_{\mathrm{CD}}(-): \mathrm{Sch}_{B}^{\mathrm{op}} \rightarrow \mathbf{T r}^{\otimes}DMCD(−):SchBop→Tr⊗, admitting a sixfunctor formalism. There are also Tate twists M↦M(n)M↦M(n)M|->M(n)M \mapsto M(n)M↦M(n). This gives a definition of motivic cohomology of an general scheme YYYYY by
which for Y∈SmSY∈SmSY inSm_(S)Y \in \mathrm{Sm}_{S}Y∈SmS agrees with the definition given above.
with ϕ∗ϕ∗phi_(**)\phi_{*}ϕ∗ playing the role of the Eilenberg-MacLane functor, giving rise to the spectrum MY∈SH(Y)MY∈SH(Y)M_(Y)inSH(Y)\mathcal{M}_{Y} \in \mathrm{SH}(Y)MY∈SH(Y) representing H∗,∗(Y,Z)H∗,∗(Y,Z)H^(**,**)(Y,Z)H^{*, *}(Y, \mathbb{Z})H∗,∗(Y,Z) [33, DEfInITION 11.2.17]. They discuss the question of whether Y↦MZYY↦MZYY|->MZ_(Y)Y \mapsto \mathcal{M} \mathbb{Z}_{Y}Y↦MZY is cartesian (see [33, CONJECTURE 11.2.22, PROPOSITION 11.4.7]), without reaching a general resolution.
Taking the base-scheme to be Spec ZZZ\mathbb{Z}Z, Spitzweck's construction yields a representing object MZZMZZMZ_(Z)M \mathbb{Z}_{\mathbb{Z}}MZZ in SH(Z)SH(Z)SH(Z)\mathrm{SH}(\mathbb{Z})SH(Z) and one can thus define absolute motivic cohomology for smooth schemes over a given base-scheme SSSSS by pulling back MZZMZZMZ_(Z)M \mathbb{Z}_{\mathbb{Z}}MZZ to MZS∈SH(S)MZS∈SH(S)MZ_(S)inSH(S)M \mathbb{Z}_{S} \in \mathrm{SH}(S)MZS∈SH(S). The resulting motivic cohomology agrees with Voevodsky's for smooth schemes of finite type over a perfect base-field, and with the hypercohomology of the Bloch cycle complex for smooth finite type schemes over a Dedekind domain. This gives rise to a triangulated category of motives DMsp(S)DMsp(S)DM_(sp)(S)\mathrm{DM}_{\mathrm{sp}}(S)DMsp(S) over a base-scheme SSSSS, defined as the homotopy category of MZS−MZS−MZ_(S^(-))M \mathbb{Z}_{S^{-}}MZS− modules, and the functor S↦DMSp(S)S↦DMSp(S)S|->DM_(Sp)(S)S \mapsto \mathrm{DM}_{\mathrm{Sp}}(S)S↦DMSp(S) inherits a Grothendieck six-functor formalism from that of S↦SH(S)S↦SH(S)S|->SH(S)S \mapsto \mathrm{SH}(S)S↦SH(S).
3.3. Hoyois' motivic cohomology
Spitzweck's construction gives a solution to the problem of constructing a triangulated category of motives over an arbitrary base, admitting a six-functor formalism and thus yielding a good theory of motivic cohomology. His construction is a bit indirect and it would be nice to have a direct construction of a representing motivic ring spectrum HZS∈SH(S)HZS∈SH(S)HZ_(S)inSH(S)H \mathbb{Z}_{S} \in \mathrm{SH}(S)HZS∈SH(S) for each base-scheme SSSSS, still satisfying the cartesian condition.
Hoyois has constructed such a theory of motivic cohomology over an arbitrary basescheme by using a recent breakthrough in our understanding of the motivic stable homotopy categories SH(S)SH(S)SH(S)\mathrm{SH}(S)SH(S). This is a new construction of SH(S)SH(S)SH(S)\mathrm{SH}(S)SH(S) more in line with Voevodsky construction of DM(k)DM(k)DM(k)\mathrm{DM}(k)DM(k). The basic idea is sketched in notes of Voevodsky [126], which were realized in a series of works by Ananyevskiy, Garkusha, Panin, Neshitov [2,4,45-48](authorship in various combinations). Building on these works, Elmanto, Hoyois, Khan, Sosnilo, and Yakerson [36-38] construct an infinity category of framed correspondences, and use the basic program of Voevodsky's construction of DM(k)DM(k)DM(k)\mathrm{DM}(k)DM(k) to realize SH(S)SH(S)SH(S)\mathrm{SH}(S)SH(S) as arising from presheaves of spectra with framed transfers, just as objects of DM(k)DM(k)DM(k)\mathrm{DM}(k)DM(k) arise from presheaves of complexes of sheaves with transfers for finite correspondences. It is not our purpose here to give a detailed discussion of this beautiful topic; we content ourselves with sketching some of the basic principles.
An integral closed subscheme Z⊂X×YZ⊂X×YZ sub X xx YZ \subset X \times YZ⊂X×Y that defines a finite correspondence from XXXXX to YYYYY can be thought of a special type of a span via the two projections
For XXXXX and YYYYY smooth and finite type over a given base-scheme SSSSS, a framed correspondence from XXXXX to YYYYY is also a span,
satisfying certain conditions, together with some additional data (the framing). For simplicity, assume that XXXXX is connected. The morphism ppppp is required to be a finite, flat, local complete intersection (lci) morphism, called a finite syntomic morphism (the terminology was introduced by Mazur). The lci condition means that ppppp factors as closed immersion i:Z→Pi:Z→Pi:Z rarr Pi: Z \rightarrow Pi:Z→P followed by a smooth morphism f:P→Xf:P→Xf:P rarr Xf: P \rightarrow Xf:P→X, and the closed subscheme i(Z)i(Z)i(Z)i(Z)i(Z) of PPPPP is locally defined by exactly dimXP−dimXZdimXâ¡P−dimXâ¡Zdim_(X)P-dim_(X)Z\operatorname{dim}_{X} P-\operatorname{dim}_{X} ZdimXâ¡P−dimXâ¡Z equations forming a regular sequence. The morphism ppppp factored in this way has a relative cotangent complex LpLpL_(p)\mathbb{L}_{p}Lp admitting a simple description, namely
A framing for a syntomic map p:Z→Xp:Z→Xp:Z rarr Xp: Z \rightarrow Xp:Z→X is a choice of a path γ:[0,1]→K(Z)γ:[0,1]→K(Z)gamma:[0,1]rarrK(Z)\gamma:[0,1] \rightarrow \mathcal{K}(Z)γ:[0,1]→K(Z) connecting {Lp}Lp{L_(p)}\left\{\mathbb{L}_{p}\right\}{Lp} with the base-point 0∈K(Z)0∈K(Z)0inK(Z)0 \in \mathcal{K}(Z)0∈K(Z). For a framing to exist, the class [Lp]∈K0(Z)Lp∈K0(Z)[L_(p)]inK_(0)(Z)\left[\mathbb{L}_{p}\right] \in K_{0}(Z)[Lp]∈K0(Z) must be zero, but the choice of γγgamma\gammaγ is additional data. The morphism q:Z→Yq:Z→Yq:Z rarr Yq: Z \rightarrow Yq:Z→Y is arbitrary.
One has the usual notion of a composition of spans:
which preserves the finite syntomic condition. However, one needs a higher categorical structure to take care of associativity constraints. The composition of paths is even trickier, since we are dealing here with actual paths, not paths up to homotopy. In the end, this produces an infinity category Corr fr(SmS)frSmS^(fr)(Sm_(S)){ }^{\mathrm{fr}}\left(\mathrm{Sm}_{S}\right)fr(SmS) of framed correspondences on smooth SSSSS-schemes, rather than a category; roughly speaking, the composition is only defined "up to homotopy and coherent higher homotopies."
Via the infinity category Corrfr(SmS)Corrfrâ¡SmSCorr^(fr)(Sm_(S))\operatorname{Corr}^{\mathrm{fr}}\left(\mathrm{Sm}_{S}\right)Corrfrâ¡(SmS), we have the infinity category of framed motivic spaces, Hfr(S)Hfr(S)H^(fr)(S)\mathbf{H}^{\mathrm{fr}}(S)Hfr(S), this being the infinity category of A1A1A^(1)\mathbb{A}^{1}A1-invariant, Nisnevich sheaves of spaces on Corrfr(SmS)CorrfrSmSCorr^(fr)(Sm_(S))\mathbf{C o r r}{ }^{\mathrm{fr}}\left(\mathrm{Sm}_{S}\right)Corrfr(SmS). There is a stable version, SHfr(S)SHfr(S)SH^(fr)(S)\mathbf{S H}^{\mathrm{fr}}(S)SHfr(S), an infinite suspension functor Σfr∞:Hfr(S)→SHfr(S)Σfr∞:Hfr(S)→SHfr(S)Sigma_(fr)^(oo):H^(fr)(S)rarrSH^(fr)(S)\Sigma_{\mathrm{fr}}^{\infty}: \mathbf{H}^{\mathrm{fr}}(S) \rightarrow \mathbf{S H}^{\mathrm{fr}}(S)Σfr∞:Hfr(S)→SHfr(S), and an equivalence of infinity categories γ∗:SHfr(S)→SH(S)γ∗:SHfr(S)→SH(S)gamma_(**):SH^(fr)(S)rarrSH(S)\gamma_{*}: \mathbf{S H}^{\mathrm{fr}}(S) \rightarrow \mathbf{S H}(S)γ∗:SHfr(S)→SH(S), where SH(S)SH(S)SH(S)\mathbf{S H}(S)SH(S) is the infinity category version of the triangulated category SH(S)SH(S)SH(S)\mathrm{SH}(S)SH(S), that is, the homotopy category of SH(S)SH(S)SH(S)\mathbf{S H}(S)SH(S) is SH(S)SHâ¡(S)SH(S)\operatorname{SH}(S)SHâ¡(S). The equivalence γ∗γ∗gamma_(**)\gamma_{*}γ∗ can be thought of as a version of the construction of infinite loop spaces from Segal's ΓΓGamma\GammaΓ-spaces, with a framed correspondence X←Z→YXâ†Z→YX larr Z rarr YX \leftarrow Z \rightarrow YXâ†Z→Y of degree nnnnn over XXXXX being viewed as a generalization of the map [n]+→[0]+[n]+→[0]+[n]_(+)rarr[0]_(+)[n]_{+} \rightarrow[0]_{+}[n]+→[0]+ in Γop Γop Gamma^("op ")\Gamma^{\text {op }}Γop .
With this background, we can give a rough idea of Hoyois' construction of the spectrum representing motivic cohomology over SSSSS in [63]. He considers spans X←pZ→qYXâ†pZ→qYXlarr^(p)Zrarr"q"YX \stackrel{p}{\leftarrow} Z \xrightarrow{q} YXâ†pZ→qY, X,Y∈SmSX,Y∈SmSX,Y inSm_(S)X, Y \in \operatorname{Sm}_{S}X,Y∈SmS, with p:Z→Xp:Z→Xp:Z rarr Xp: Z \rightarrow Xp:Z→X a finite morphism such that p∗OZp∗OZp_(**)O_(Z)p_{*} \mathcal{O}_{Z}p∗OZ is a locally free OX−OX−O_(X^(-))\mathcal{O}_{X^{-}}OX− module; note that this condition is satisfied if ppppp is a syntomic morphism, but not conversely. These spans form a category Corr flf (SmS)flf SmS^("flf ")(Sm_(S)){ }^{\text {flf }}\left(\mathrm{Sm}_{S}\right)flf (SmS) under span composition ("flf" stands for "finite, locally free") and forgetting the paths γγgamma\gammaγ defines a morphism of (infinity) categories πad:Corrfr(SmS)→Corrflf(SmS)Ï€ad:Corrfrâ¡SmS→Corrflfâ¡SmSpi_(ad):Corr^(fr)(Sm_(S))rarrCorr^(flf)(Sm_(S))\pi_{\mathrm{ad}}: \operatorname{Corr}^{\mathrm{fr}}\left(\mathrm{Sm}_{S}\right) \rightarrow \operatorname{Corr}^{\mathrm{flf}}\left(\mathrm{Sm}_{S}\right)Ï€ad:Corrfrâ¡(SmS)→Corrflfâ¡(SmS).
Given a commutative monoid AAAAA, the constant Nisnevich sheaf on SmSSmSSm_(S)\mathrm{Sm}_{S}SmS with value AAAAA extends to a functor
AS:(Corrflf)op→AbAS:Corrflfop→AbA_(S):(Corr^(flf))^(op)rarrAbA_{S}:\left(\mathbf{C o r r}^{\mathrm{flf}}\right)^{\mathrm{op}} \rightarrow \mathbf{A b}AS:(Corrflf)op→Ab
where the pullback from YYYYY to XXXXX by X←pZ→qYXâ†pZ→qYXlarr^(p)Zrarr"q"YX \stackrel{p}{\leftarrow} Z \xrightarrow{q} YXâ†pZ→qY is given by multiplication by rnkOXOZrnkOXOZrnk_(O_(X))O_(Z)\mathrm{rnk}_{\mathcal{O}_{X}} \mathcal{O}_{Z}rnkOXOZ if XXXXX and YYYYY are connected; one extends to general smooth XXXXX and YYYYY by additivity. This gives us the presheaf (of abelian monoids) with framed transfers ASfr:=AS∘πadopASfr:=AS∘πadopA_(S)^(fr):=A_(S)@pi_(ad)^(op)A_{S}^{\mathrm{fr}}:=A_{S} \circ \pi_{\mathrm{ad}}^{\mathrm{op}}ASfr:=AS∘πadop, and the machinery of [36-38] converts this into the motivic spectrum γ∗Σfr∞ASfr∈SH(S)γ∗Σfr∞ASfr∈SHâ¡(S)gamma_(**)Sigma_(fr)^(oo)A_(S)^(fr)in SH(S)\gamma_{*} \Sigma_{\mathrm{fr}}^{\infty} A_{S}^{\mathrm{fr}} \in \operatorname{SH}(S)γ∗Σfr∞ASfr∈SHâ¡(S). Hoyois shows [63,
LEMMA 20] that this construction produces a cartesian family, and that taking A=ZA=ZA=ZA=\mathbb{Z}A=Z recovers Spitzweck's family S↦MZSS↦MZSS|->MZ_(S)S \mapsto M \mathbb{Z}_{S}S↦MZS [63, THEOREM 21].
This gives us a conceptually simple construction of a motivic Eilenberg-MacLane spectrum, and the corresponding motivic category DMH(S)DMH(S)DM_(H)(S)\mathrm{DM}_{H}(S)DMH(S), much in the spirit of Voevodsky original construction of DM(k)DM(k)DM(k)\mathrm{DM}(k)DM(k) and the Röndigs- stvær theorem identifying DM(k)DM(k)DM(k)\mathrm{DM}(k)DM(k) with the homotopy category of EM(Z(0))EMâ¡(Z(0))EM(Z(0))\operatorname{EM}(\mathbb{Z}(0))EMâ¡(Z(0))-modules.
4. MILNOR-WITT MOTIVIC COHOMOLOGY
The classical Chow group CHn(X)CHn(X)CH^(n)(X)\mathrm{CH}^{n}(X)CHn(X) of codimension nnnnn algebraic cycles modulo rational equivalence on a smooth variety XXXXX is part of the motivic cohomology of XXXXX via the isomorphism CHn(X)=H2n(X,Z(n))CHn(X)=H2n(X,Z(n))CH^(n)(X)=H^(2n)(X,Z(n))\mathrm{CH}^{n}(X)=H^{2 n}(X, \mathbb{Z}(n))CHn(X)=H2n(X,Z(n)). Barge and Morel [12] have introduced a refinement of the Chow groups, the Chow-Witt groups, that incorporates information about quadratic forms. Their construction has been embedded in a larger theory of Milnor-Witt motives and Milnor-Witt motivic cohomology, which we describe in this section. The quadratic information given by the Chow-Witt groups, Milnor-Witt motivic cohomology and related theories has proven useful in recent efforts to give quadratic refinements for intersection theory and enumerative geometry; see [10,11,21,61,76,77,86][10,11,21,61,76,77,86][10,11,21,61,76,77,86][10,11,21,61,76,77,86][10,11,21,61,76,77,86] for some examples. We refer the reader to [8,31,39,92][8,31,39,92][8,31,39,92][8,31,39,92][8,31,39,92] for details on the theory described in this section.
4.1. Milnor-Witt KKKKK-theory and the Chow-Witt groups
A codimension nnnnn algebraic cycle Z:=∑iniZiZ:=∑i niZiZ:=sum_(i)n_(i)Z_(i)Z:=\sum_{i} n_{i} Z_{i}Z:=∑iniZi can be thought of as the set of its generic points ziziz_(i)z_{i}zi together with the ZZZ\mathbb{Z}Z-valued function ninin_(i)n_{i}ni on ziziz_(i)z_{i}zi, from which we can write the group Zn(X)Zn(X)Z^(n)(X)Z^{n}(X)Zn(X) of codimension nnnnn algebraic cycles as
where X(n)X(n)X^((n))X^{(n)}X(n) is the set of points z∈Xz∈Xz in Xz \in Xz∈X with closure Z:=z¯⊂XZ:=z¯⊂XZ:= bar(z)sub XZ:=\bar{z} \subset XZ:=z¯⊂X of codimension nnnnn.
Let GW(F)GW(F)GW(F)\mathrm{GW}(F)GW(F) denote the Grothendieck-Witt ring of virtual non-degenerate quadratic forms over FFFFF and let W(F)=GW(F)/(H)W(F)=GWâ¡(F)/(H)W(F)=GW(F)//(H)W(F)=\operatorname{GW}(F) /(H)W(F)=GWâ¡(F)/(H) where HHHHH is the hyperbolic form H(x,y)=H(x,y)=H(x,y)=H(x, y)=H(x,y)=x2−y2x2−y2x^(2)-y^(2)x^{2}-y^{2}x2−y2 (we assume throughout that the characteristic is ≠2≠2!=2\neq 2≠2 to avoid technical difficulties); W(F)W(F)W(F)W(F)W(F) is the Witt ring of anisotropic quadratic forms over FFFFF (see [107]).
One can consider a finite set of codimension nnnnn points zi∈X(n)zi∈X(n)z_(i)inX^((n))z_{i} \in X^{(n)}zi∈X(n), together with a collection of classes {qi∈GW(k(zi))}qi∈GWkzi{q_(i)inGW(k(z_(i)))}\left\{q_{i} \in \mathrm{GW}\left(k\left(z_{i}\right)\right)\right\}{qi∈GW(k(zi))}; one recovers a ZZZ\mathbb{Z}Z-valued function on ziziz_(i)z_{i}zi by taking the rank of qiqiq_(i)q_{i}qi. This gives the group
with rank homomorphism rnk: Z~n(X)→Zn(X)Z~n(X)→Zn(X)tilde(Z)^(n)(X)rarrZ^(n)(X)\tilde{Z}^{n}(X) \rightarrow Z^{n}(X)Z~n(X)→Zn(X). In contrast with integer-valued functions, an element q∈GW(k(z))q∈GW(k(z))q inGW(k(z))q \in \mathrm{GW}(k(z))q∈GW(k(z)) does not always extend to all of z¯z¯bar(z)\bar{z}z¯; there is an obstruction given by a certain boundary map
This starts to look more like classical homology, in that one should consider Z~n(X)Z~n(X)tilde(Z)^(n)(X)\tilde{Z}^{n}(X)Z~n(X) as a group of chains rather than a group of cycles.
This is not enough, as one needs a quadratic refinement for the classical relation given by rational equivalence. The original construction of Barge-Morel defined this relation, but later developments put their construction in a rather more natural form, which we now describe.
We recall that the Milnor KKKKK-theory ring K∗M(F):=⨁n≥0KnM(F)K∗M(F):=â¨n≥0 KnM(F)K_(**)^(M)(F):=bigoplus_(n >= 0)K_(n)^(M)(F)K_{*}^{M}(F):=\bigoplus_{n \geq 0} K_{n}^{M}(F)K∗M(F):=â¨n≥0KnM(F) of a field FFFFF is defined as the quotient of the tensor algebra on the abelian group of units F×F×F^(xx)F^{\times}F×, modulo the Steinberg relation
The quadratic refinement of K∗M(F)K∗M(F)K_(**)^(M)(F)K_{*}^{M}(F)K∗M(F) is the Hopkins-Morel Milnor-Witt KKKKK-theory of FFFFF.
Definition 4.1 (Hopkins-Morel [92, Definition 6.3.1]). Let FFFFF be a field. The Milnor-Witt KKKKK-theory of F,K∗MW(F):=⨁n∈ZKnMW(F)F,K∗MW(F):=â¨n∈Z KnMW(F)F,K_(**)^(MW)(F):=bigoplus_(n inZ)K_(n)^(MW)(F)F, K_{*}^{\mathrm{MW}}(F):=\bigoplus_{n \in \mathbb{Z}} K_{n}^{\mathrm{MW}}(F)F,K∗MW(F):=â¨n∈ZKnMW(F), is the ZZZ\mathbb{Z}Z-graded associative algebra defined by the following generators and relations.
Generators
(G1) For each u∈F×u∈F×u inF^(xx)u \in F^{\times}u∈F×, we have the generator [u][u][u][u][u] of degree 1 ;
(G2) There is an additional generator ηηeta\etaη of degree -1 .
(R2) [u]⋅[1−u]=0[u]â‹…[1−u]=0[u]*[1-u]=0[u] \cdot[1-u]=0[u]â‹…[1−u]=0 for u∈F∖{0,1}u∈F∖{0,1}u in F\\{0,1}u \in F \backslash\{0,1\}u∈F∖{0,1};
(R3) Let h=(2+η⋅[−1])h=(2+η⋅[−1])h=(2+eta*[-1])h=(2+\eta \cdot[-1])h=(2+η⋅[−1]). Then η⋅h=0η⋅h=0eta*h=0\eta \cdot h=0η⋅h=0.
It follows directly that sending [u][u][u][u][u] to {u}∈K1M(F){u}∈K1M(F){u}inK_(1)^(M)(F)\{u\} \in K_{1}^{M}(F){u}∈K1M(F) and sending ηηeta\etaη to zero defines a surjective graded algebra homomorphism K∗MW(F)→K∗M(F)K∗MW(F)→K∗M(F)K_(**)^(MW)(F)rarrK_(**)^(M)(F)K_{*}^{M W}(F) \rightarrow K_{*}^{M}(F)K∗MW(F)→K∗M(F) with kernel (η)(η)(eta)(\eta)(η). We write [u1,…,un]u1,…,un[u_(1),dots,u_(n)]\left[u_{1}, \ldots, u_{n}\right][u1,…,un] for the product [u1]⋯[un]u1⋯un[u_(1)]cdots[u_(n)]\left[u_{1}\right] \cdots\left[u_{n}\right][u1]⋯[un].
Theorem 4.2 (Hopkins-Morel [92, THEOREM 6.4.5]). Let I(F)⊂GW(F)I(F)⊂GW(F)I(F)subGW(F)I(F) \subset \mathrm{GW}(F)I(F)⊂GW(F) be the kernel of the rank homomorphism GW(F)→ZGW(F)→ZGW(F)rarrZ\mathrm{GW}(F) \rightarrow \mathbb{Z}GW(F)→Z, with the nth power ideal In(F)⊂GW(F)In(F)⊂GW(F)I^(n)(F)subGW(F)I^{n}(F) \subset \mathrm{GW}(F)In(F)⊂GW(F) for n>0n>0n > 0n>0n>0. Define In(F)=W(F)In(F)=W(F)I^(n)(F)=W(F)I^{n}(F)=W(F)In(F)=W(F) for n≤0n≤0n <= 0n \leq 0n≤0. Then for each n∈Zn∈Zn inZn \in \mathbb{Z}n∈Z, the surjection KnMW(F)→KnM(F)KnMW(F)→KnM(F)K_(n)^(MW)(F)rarrK_(n)^(M)(F)K_{n}^{\mathrm{MW}}(F) \rightarrow K_{n}^{M}(F)KnMW(F)→KnM(F) extends to an exact sequence
For n=0,K0M(F)=Z,K0MW(F)n=0,K0M(F)=Z,K0MW(F)n=0,K_(0)^(M)(F)=Z,K_(0)^(MW)(F)n=0, K_{0}^{M}(F)=\mathbb{Z}, K_{0}^{\mathrm{MW}}(F)n=0,K0M(F)=Z,K0MW(F) is isomorphic to GW(F)GW(F)GW(F)\mathrm{GW}(F)GW(F) and the above sequence is isomorphic to the defining sequence for I(F)I(F)I(F)I(F)I(F). For n<0,KnM(F)=0n<0,KnM(F)=0n < 0,K_(n)^(M)(F)=0n<0, K_{n}^{M}(F)=0n<0,KnM(F)=0 and KnMW(F)≅KnMW(F)≅K_(n)^(MW)(F)~=K_{n}^{\mathrm{MW}}(F) \congKnMW(F)≅ W(F)W(F)W(F)W(F)W(F). Finally, we have, for each n<0n<0n < 0n<0n<0, a commutative diagram
Given a dvrOdvrâ¡Odvr O\operatorname{dvr} \mathcal{O}dvrâ¡O with residue field kkkkk, quotient field FFFFF, and generator ttttt for the maximal ideal, one has the map
for u1,…,un∈O×u1,…,un∈O×u_(1),dots,u_(n)inO^(xx)u_{1}, \ldots, u_{n} \in \mathcal{O}^{\times}u1,…,un∈O×, and x∈Kn+1MW(F)x∈Kn+1MW(F)x inK_(n+1)^(MW)(F)x \in K_{n+1}^{\mathrm{MW}}(F)x∈Kn+1MW(F), where u¯iu¯ibar(u)_(i)\bar{u}_{i}u¯i is the image of uiuiu_(i)u_{i}ui in k×k×k^(xx)k^{\times}k×. This is similar to the well-known boundary map ∂:KnM(F)→Kn−1M(k)∂:KnM(F)→Kn−1M(k)del:K_(n)^(M)(F)rarrK_(n-1)^(M)(k)\partial: K_{n}^{M}(F) \rightarrow K_{n-1}^{M}(k)∂:KnM(F)→Kn−1M(k), with the difference, that ∂∂del\partial∂ does not depend on the choice of ttttt while ∂t∂tdel_(t)\partial_{t}∂t does. To get a boundary map that is independent of the choice of parameter ttttt, one needs to include the twisting. This yields the well-defined boundary map
∂:KnMW(F;L⊗OF)→Kn−1MW(k;L⊗O(m/m2)∨)∂:KnMWF;L⊗OF→Kn−1MWk;L⊗Om/m2∨del:K_(n)^(MW)(F;Lox_(O)F)rarrK_(n-1)^(MW)(k;Lox_(O)(m//m^(2))^(vv))\partial: K_{n}^{\mathrm{MW}}\left(F ; L \otimes_{\mathcal{O}} F\right) \rightarrow K_{n-1}^{\mathrm{MW}}\left(k ; L \otimes_{\mathcal{O}}\left(\mathfrak{m} / \mathfrak{m}^{2}\right)^{\vee}\right)∂:KnMW(F;L⊗OF)→Kn−1MW(k;L⊗O(m/m2)∨)
for LLLLL a free rank-one OOO\mathcal{O}O-module, independent of the choice of generator for the maximal ideal mmm\mathfrak{m}m, where ∂∂del\partial∂ is defined by choosing a generator ttttt and an OOO\mathcal{O}O-basis λλlambda\lambdaλ for LLLLL, and setting
Definition 4.3. Let XXXXX be a smooth finite type kkkkk-scheme, and let LLL\mathscr{L}L be an invertible sheaf on XXXXX. The nnnnnth LLL\mathscr{L}L-twisted Rost-Schmid complex for Milnor-Witt KKKKK-theory is the complex RS∗(X,L,n)RS∗â¡(X,L,n)RS^(**)(X,L,n)\operatorname{RS}^{*}(X, \mathscr{L}, n)RS∗â¡(X,L,n) with
and boundary map ∂m:RSm(X,L,n)→RSm+1(X,L,n)∂m:RSmâ¡(X,L,n)→RSm+1â¡(X,L,n)del^(m):RS^(m)(X,L,n)rarrRS^(m+1)(X,L,n)\partial^{m}: \operatorname{RS}^{m}(X, \mathscr{L}, n) \rightarrow \operatorname{RS}^{m+1}(X, \mathscr{L}, n)∂m:RSmâ¡(X,L,n)→RSm+1â¡(X,L,n) the sum of the maps
The twisted Milnor-Witt sheaf KnMW(L)XKnMW(L)XK_(n)^(MW)(L)_(X)\mathcal{K}_{n}^{\mathrm{MW}}(\mathscr{L})_{X}KnMW(L)X is the Nisnevich sheaf on XXXXX associated to the presheaf
The codimension nnnnn twisted Chow-Witt group of X,CHn~(X;L)X,CHn~(X;L)X, widetilde(CH^(n))(X;L)X, \widetilde{\mathrm{CH}^{n}}(X ; \mathscr{L})X,CHn~(X;L), is defined as
with essentially the same definition as the Rost-Schmid complex, without the twisting. This gives us the Milnor KKKKK-theory sheaf Kn,XM:=ker∂0Kn,XM:=kerâ¡âˆ‚0K_(n,X)^(M):=ker del^(0)\mathcal{K}_{n, X}^{M}:=\operatorname{ker} \partial^{0}Kn,XM:=kerâ¡âˆ‚0, and it follows easily from the definitions that CHn(X)=Hn(G∗(X,n))CHn(X)=HnG∗(X,n)CH^(n)(X)=H^(n)(G^(**)(X,n))\mathrm{CH}^{n}(X)=H^{n}\left(G^{*}(X, n)\right)CHn(X)=Hn(G∗(X,n)). The same ideas that lead to the Bloch-Kato formula [78]
(see the discussion following [31, DEFINITION 3.1] for details). The maps KnMW→KnMKnMW→KnMK_(n)^(MW)rarrK_(n)^(M)\mathcal{K}_{n}^{\mathrm{MW}} \rightarrow \mathcal{K}_{n}^{\mathrm{M}}KnMW→KnM give the map of complexes RS∗(X,L,n)→G∗(X,n)RS∗â¡(X,L,n)→G∗(X,n)RS^(**)(X,L,n)rarrG^(**)(X,n)\operatorname{RS}^{*}(X, \mathscr{L}, n) \rightarrow G^{*}(X, n)RS∗â¡(X,L,n)→G∗(X,n) and the corresponding map rnk X,nX,n_(X,n){ }_{X, n}X,n : CH~n(X;L)→CHn(X)CH~n(X;L)→CHn(X)widetilde(CH)^(n)(X;L)rarrCH^(n)(X)\widetilde{\mathrm{CH}}^{n}(X ; \mathscr{L}) \rightarrow \mathrm{CH}^{n}(X)CH~n(X;L)→CHn(X).
The twists by an invertible sheaf are not just a device for defining the Rost-Schmid complexes and the Chow-Witt groups, they play an integral part in the structure of the overall theory. The Chow groups of smooth varieties admit the functorialities of a Borel-Moore homology theory: they have functorial pullback maps f∗:CHn(Y)→CHn(X)f∗:CHn(Y)→CHn(X)f^(**):CH^(n)(Y)rarrCH^(n)(X)f^{*}: \mathrm{CH}^{n}(Y) \rightarrow \mathrm{CH}^{n}(X)f∗:CHn(Y)→CHn(X) for each morphism f:X→Yf:X→Yf:X rarr Yf: X \rightarrow Yf:X→Y in SmkSmkSm_(k)\operatorname{Sm}_{k}Smk, and for f:X→Yf:X→Yf:X rarr Yf: X \rightarrow Yf:X→Y a proper morphism of relative dimension ddddd, one has the functorial proper push-forward map f∗:CHn(X)→CHn−d(Y)f∗:CHn(X)→CHn−d(Y)f_(**):CH^(n)(X)rarrCH^(n-d)(Y)f_{*}: \mathrm{CH}^{n}(X) \rightarrow \mathrm{CH}^{n-d}(Y)f∗:CHn(X)→CHn−d(Y). The ChowWitt groups also have a contravariant functoriality; for f:X→Yf:X→Yf:X rarr Yf: X \rightarrow Yf:X→Y, and LLL\mathscr{L}L an invertible sheaf on YYYYY, one has the functorial pullback
This limits the possible twists CH~n(X,M)CH~n(X,M)widetilde(CH)^(n)(X,M)\widetilde{\mathrm{CH}}^{n}(X, \mathcal{M})CH~n(X,M) for which a push-forward f∗f∗f_(**)f_{*}f∗ is even defined; this type of restricted push-forward is typical of so-called SL-oriented theories, such as hermitian KKKKK-theory. See [1] for a detailed discussion of SL-oriented theories and [31, chAP. 3] for the details concerning the push-forward in CH~∗CH~∗widetilde(CH)^(**)\widetilde{\mathrm{CH}}^{*}CH~∗.
4.2. The homotopy ttttt-structure and Morel's theorem
Building on the Bloch-Kato formula, CHn(X)≅Hn(XNis,Kn,XM)CHn(X)≅HnXNis,Kn,XMCH^(n)(X)~=H^(n)(X_(Nis),K_(n,X)^(M))\mathrm{CH}^{n}(X) \cong H^{n}\left(X_{\mathrm{Nis}}, \mathcal{K}_{n, X}^{M}\right)CHn(X)≅Hn(XNis,Kn,XM), one can construct a good bigraded cohomology theory EM(K∗M)∗∗EMâ¡K∗M∗∗EM (K_(**)^(M))^(****)\operatorname{EM}\left(\mathcal{K}_{*}^{M}\right)^{* *}EMâ¡(K∗M)∗∗ by using all the cohomology groups. To get the correct bigrading, one should set
giving in particular EM(K∗M)2n,n(X)=CHn(X)EMâ¡K∗M2n,n(X)=CHn(X)EM (K_(**)^(M))^(2n,n)(X)=CH^(n)(X)\operatorname{EM}\left(\mathcal{K}_{*}^{M}\right)^{2 n, n}(X)=\mathrm{CH}^{n}(X)EMâ¡(K∗M)2n,n(X)=CHn(X). It was recognized early on that this theory is not the sought-after motivic cohomology, for instance, for X=SpecF,FX=Specâ¡F,FX=Spec F,FX=\operatorname{Spec} F, FX=Specâ¡F,F a field, one gets exactly the Milnor KKKKK-theory of FFFFF, and none of the other parts of the KKKKK-theory of FFFFF. In spite of this, this theory and the similarly defined theory for Milnor-Witt KKKKK-theory have a natural place in the universe of motivic cohomology theories, which we now explain.
The classical stable homotopy category SHSHSH\mathrm{SH}SH is a triangulated category with a natural ttttt-structure measuring connectedness, mentioned in Section 2.5. For SHSHSH\mathrm{SH}SH, the truncations give the terms in the Moore-Postnikov tower
with τ≥nE→Eτ≥nE→Etau_( >= n)E rarr E\tau_{\geq n} E \rightarrow Eτ≥nE→E characterized by inducing an isomorphism on πmÏ€mpi_(m)\pi_{m}Ï€m for m≥nm≥nm >= nm \geq nm≥n and with πmτ≥nE=0Ï€mτ≥nE=0pi_(m)tau_( >= n)E=0\pi_{m} \tau_{\geq n} E=0Ï€mτ≥nE=0 for m<nm<nm < nm<nm<n. The heart of SH is the category of spectra EEEEE with πmE=0Ï€mE=0pi_(m)E=0\pi_{m} E=0Ï€mE=0 for m≠0m≠0m!=0m \neq 0m≠0, which are just the Eilenberg-MacLane spectra EM(A),AEMâ¡(A),AEM(A),A\operatorname{EM}(A), AEMâ¡(A),A an abelian group. Thus, the heart of SHSHSH\mathrm{SH}SH is AbAbAb\mathbf{A b}Ab and the cohomology theory represented by τ0EÏ„0Etau_(0)E\tau_{0} EÏ„0E is
singular cohomology with coefficients in the abelian group π0EÏ€0Epi_(0)E\pi_{0} EÏ€0E.
We have a parallel ttttt-structure on SH(k)SH(k)SH(k)\mathrm{SH}(k)SH(k), introduced by Morel [92, §5.2], called the homotopy ttttt-structure (and not coming from Voevodsky's slice tower discussed in Section 2.5). This is similar to the ttttt-structure on SHSHSH\mathrm{SH}SH, where one takes into account the fact that one has bigraded homotopy sheaves πa,bEÏ€a,bEpi_(a,b)E\pi_{a, b} EÏ€a,bE for E∈SH(k)E∈SH(k)E inSH(k)E \in \mathrm{SH}(k)E∈SH(k), rather than a ZZZ\mathbb{Z}Z-graded family of homotopy groups πnEÏ€nEpi_(n)E\pi_{n} EÏ€nE for E∈SHE∈SHE inSHE \in \mathrm{SH}E∈SH. The truncation τ≥nEτ≥nEtau_( >= n)E\tau_{\geq n} Eτ≥nE is characterized by
πa,b(τ≥nE)={πa,b(E) if a−b≥n0 if a−b<nÏ€a,bτ≥nE=Ï€a,b(E) if a−b≥n0 if a−b<npi_(a,b)(tau_( >= n)E)={[pi_(a,b)(E)," if "a-b >= n],[0," if "a-b < n]:}\pi_{a, b}\left(\tau_{\geq n} E\right)= \begin{cases}\pi_{a, b}(E) & \text { if } a-b \geq n \\ 0 & \text { if } a-b<n\end{cases}Ï€a,b(τ≥nE)={Ï€a,b(E) if a−b≥n0 if a−b<n
Recalling that the sphere Sa,bSa,bS^(a,b)S^{a, b}Sa,b is Sa−b∧GmbSa−b∧GmbS^(a-b)^^G_(m)^(b)S^{a-b} \wedge \mathbb{G}_{m}^{b}Sa−b∧Gmb, the homotopy ttttt-structure on SH(k)SH(k)SH(k)\mathrm{SH}(k)SH(k) is measuring S1S1S^(1)S^{1}S1-connectedness, instead of the P1P1P^(1)\mathbb{P}^{1}P1-connectedness measured by Voevodsky's slice tower.
We denote the 0 th truncation τ0EÏ„0Etau_(0)E\tau_{0} EÏ„0E for E∈SH(k)E∈SH(k)E inSH(k)E \in \mathrm{SH}(k)E∈SH(k) by EM(π−∗,−∗E)EMâ¡Ï€âˆ’∗,−∗EEM(pi_(-**,-**)E)\operatorname{EM}\left(\pi_{-*,-*} E\right)EMâ¡(π−∗,−∗E); the notation comes from Morel's identification of the heart with his category of homotopy modules; for details, see [92, §5.2]. The corresponding cohomology theory satisfies, for X∈SmkX∈SmkX inSm_(k)X \in \mathrm{Sm}_{k}X∈Smk,
Here we have Morel's fundamental theorem [92, THEOREM 6.4.1] computing τ0Ï„0tau_(0)\tau_{0}Ï„0 of the sphere spectrum 1k∈SH(k)1k∈SH(k)1_(k)inSH(k)1_{k} \in \mathrm{SH}(k)1k∈SH(k).
Theorem 4.4 (Morel). Let kkkkk be a perfect field. Then there are canonical isomorphisms of sheaves on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk
s01k≅HZs01k≅HZs_(0)1_(k)~=HZs_{0} 1_{k} \cong H \mathbb{Z}s01k≅HZ
of [9,85,122][9,85,122][9,85,122][9,85,122][9,85,122], we have
Theorem 4.5. Let kkkkk be a perfect field. Then
τ0s01k=τ0HZ=EM(K∗M)Ï„0s01k=Ï„0HZ=EMâ¡K∗Mtau_(0)s_(0)1_(k)=tau_(0)HZ=EM(K_(**)^(M))\tau_{0} s_{0} 1_{k}=\tau_{0} H \mathbb{Z}=\operatorname{EM}\left(\mathcal{K}_{*}^{M}\right)Ï„0s01k=Ï„0HZ=EMâ¡(K∗M)
for X∈SmkX∈SmkX inSm_(k)X \in \operatorname{Sm}_{k}X∈Smk.
Bachmann proves an extension of this result. Recall Voevodsky's slice tower
⋯→fn+1E→fnE→⋯→f0E→⋯→E⋯→fn+1E→fnE→⋯→f0E→⋯→Ecdots rarrf_(n+1)E rarrf_(n)E rarr cdots rarrf_(0)E rarr cdots rarr E\cdots \rightarrow f_{n+1} E \rightarrow f_{n} E \rightarrow \cdots \rightarrow f_{0} E \rightarrow \cdots \rightarrow E⋯→fn+1E→fnE→⋯→f0E→⋯→E
with snEsnEs_(n)Es_{n} EsnE the layer given by the distinguished triangle
fn+1E→fnE→snE→fn+1E[1]fn+1E→fnE→snE→fn+1E[1]f_(n+1)E rarrf_(n)E rarrs_(n)E rarrf_(n+1)E[1]f_{n+1} E \rightarrow f_{n} E \rightarrow s_{n} E \rightarrow f_{n+1} E[1]fn+1E→fnE→snE→fn+1E[1]
Recall that this is not the truncation tower of a ttttt-structure, as the subcategories defined by the layers sn:=fn/fn+1sn:=fn/fn+1s_(n):=f_(n)//f_(n+1)s_{n}:=f_{n} / f_{n+1}sn:=fn/fn+1 are triangulated categories, not abelian categories.
Proposition 4.6 ([7, LEMMA 12]). Let 1k→EM(K∗M)1k→EMâ¡K∗M1_(k)rarr EM(K_(**)^(M))1_{k} \rightarrow \operatorname{EM}\left(\mathcal{K}_{*}^{M}\right)1k→EMâ¡(K∗M) be the composition 1k→τ01k=1k→τ01k=1_(k)rarrtau_(0)1_(k)=1_{k} \rightarrow \tau_{0} 1_{k}=1k→τ01k=EM(K∗MW)→EM(K∗M)EMâ¡K∗MW→EMâ¡K∗MEM(K_(**)^(MW))rarr EM(K_(**)^(M))\operatorname{EM}\left(\mathcal{K}_{*}^{\mathrm{MW}}\right) \rightarrow \operatorname{EM}\left(\mathcal{K}_{*}^{M}\right)EMâ¡(K∗MW)→EMâ¡(K∗M), the latter map induced by the surjection K∗MW→K∗MK∗MW→K∗MK_(**)^(MW)rarrK_(**)^(M)\mathcal{K}_{*}^{\mathrm{MW}} \rightarrow \mathcal{K}_{*}^{M}K∗MW→K∗M. Then the induced maps
s0(1k)→s0EM(K∗M)←f0EM(K∗M)=f0τ0HZs01k→s0EMâ¡K∗Mâ†f0EMâ¡K∗M=f0Ï„0HZs_(0)(1_(k))rarrs_(0)EM(K_(**)^(M))larrf_(0)EM(K_(**)^(M))=f_(0)tau_(0)HZs_{0}\left(1_{k}\right) \rightarrow s_{0} \operatorname{EM}\left(\mathcal{K}_{*}^{M}\right) \leftarrow f_{0} \operatorname{EM}\left(\mathcal{K}_{*}^{M}\right)=f_{0} \tau_{0} H \mathbb{Z}s0(1k)→s0EMâ¡(K∗M)â†f0EMâ¡(K∗M)=f0Ï„0HZ
are all isomorphisms, so all of these objects are isomorphic to the motivic cohomology spectrum HZHZHZH \mathbb{Z}HZ.
The truncation functors for the homotopy ttttt-structure and for the Voevodsky slice tower do not commute. Since 1k1k1_(k)1_{k}1k is effective, we have f01k=1kf01k=1kf_(0)1_(k)=1_(k)f_{0} 1_{k}=1_{k}f01k=1k and so τ0f01k=τ01k=Ï„0f01k=Ï„01k=tau_(0)f_(0)1_(k)=tau_(0)1_(k)=\tau_{0} f_{0} 1_{k}=\tau_{0} 1_{k}=Ï„0f01k=Ï„01k=EM(K∗MW)EMâ¡K∗MWEM(K_(**)^(MW))\operatorname{EM}\left(\mathcal{K}_{*}^{\mathrm{MW}}\right)EMâ¡(K∗MW). The truncations in the other order give us something new.
4.3. Milnor-Witt motivic cohomology
Definition 4.7 ([7, notation, P. 1134, JUSt BEFORE LEMMA 12]). Let kkkkk be a perfect field. Define the Milnor-Witt motivic cohomology spectrum H~Z∈SH(k)eff H~Z∈SH(k)eff tilde(H)ZinSH(k)^("eff ")\tilde{H} \mathbb{Z} \in \mathrm{SH}(k)^{\text {eff }}H~Z∈SH(k)eff by
The canonical map τ01k→τ0s01k=τ0HZÏ„01k→τ0s01k=Ï„0HZtau_(0)1_(k)rarrtau_(0)s_(0)1_(k)=tau_(0)HZ\tau_{0} 1_{k} \rightarrow \tau_{0} s_{0} 1_{k}=\tau_{0} H \mathbb{Z}Ï„01k→τ0s01k=Ï„0HZ induces the map
H~Z=f0(τ01k)→Ξf0τ0HZ=HZ.H~Z=f0Ï„01k→Ξf0Ï„0HZ=HZ.tilde(H)Z=f_(0)(tau_(0)1_(k))rarr"Xi"f_(0)tau_(0)HZ=HZ.\tilde{H} \mathbb{Z}=f_{0}\left(\tau_{0} 1_{k}\right) \xrightarrow{\Xi} f_{0} \tau_{0} H \mathbb{Z}=H \mathbb{Z} .H~Z=f0(Ï„01k)→Ξf0Ï„0HZ=HZ.
For X∈SmkX∈SmkX inSm_(k)X \in \operatorname{Sm}_{k}X∈Smk, the Milnor-Witt motivic cohomology in bidegree (a,b)(a,b)(a,b)(a, b)(a,b) is defined as H~Za,b(X)H~Za,b(X)tilde(H)Z^(a,b)(X)\tilde{H} \mathbb{Z}^{a, b}(X)H~Za,b(X).
Remarkably, one can compute H~Za,b(X)H~Za,b(X)tilde(H)Z^(a,b)(X)\tilde{H} \mathbb{Z}^{a, b}(X)H~Za,b(X) in terms of the Milnor-Witt sheaves, at least for some of the indices (a,b)(a,b)(a,b)(a, b)(a,b); one also recovers the Chow-Witt groups. For X=SpecFX=Specâ¡FX=Spec FX=\operatorname{Spec} FX=Specâ¡F, the spectrum of a field FFFFF, one has a complete computation in terms of the Milnor-Witt KKKKK-groups and the usual motivic cohomology HZa,b(X):=Ha(X,Z(b))HZa,b(X):=Ha(X,Z(b))HZ^(a,b)(X):=H^(a)(X,Z(b))H \mathbb{Z}^{a, b}(X):=H^{a}(X, \mathbb{Z}(b))HZa,b(X):=Ha(X,Z(b))
Theorem 4.8 (Bachmann). For X∈SmkX∈SmkX inSm_(k)X \in \operatorname{Sm}_{k}X∈Smk and b≤0b≤0b <= 0b \leq 0b≤0, there are natural isomorphisms
H~Za,b(X)≅Ha−b(XNis,Kb,XMW)={Ha−b(XNis,WX) for b<0Ha−b(XNis,EWX) for b=0H~Za,b(X)≅Ha−bXNis,Kb,XMW=Ha−bXNis,WX for b<0Ha−bXNis,EWX for b=0tilde(H)Z^(a,b)(X)~=H^(a-b)(X_(Nis),K_(b,X)^(MW))={[H^(a-b)(X_(Nis),W_(X))," for "b < 0],[H^(a-b)(X_(Nis),EW_(X))," for "b=0]:}\tilde{H} \mathbb{Z}^{a, b}(X) \cong H^{a-b}\left(X_{\mathrm{Nis}}, \mathcal{K}_{b, X}^{\mathrm{MW}}\right)= \begin{cases}H^{a-b}\left(X_{\mathrm{Nis}}, \mathcal{W}_{X}\right) & \text { for } b<0 \\ H^{a-b}\left(X_{\mathrm{Nis}}, \mathcal{E} \mathcal{W}_{X}\right) & \text { for } b=0\end{cases}H~Za,b(X)≅Ha−b(XNis,Kb,XMW)={Ha−b(XNis,WX) for b<0Ha−b(XNis,EWX) for b=0
Here WXWXW_(X)\mathcal{W}_{X}WX is the sheaf of Witt groups and EWXEWXEW_(X)\mathscr{E} \mathcal{W}_{X}EWX is the sheaf of Grothendieck-Witt rings.
For X∈SmkX∈SmkX inSm_(k)X \in \mathrm{Sm}_{k}X∈Smk and n∈Zn∈Zn inZn \in \mathbb{Z}n∈Z, we have
H~Z2n,n(X)≅CH~n(X)H~Z2n,n(X)≅CH~n(X)tilde(H)Z^(2n,n)(X)~= widetilde(CH)^(n)(X)\tilde{H} \mathbb{Z}^{2 n, n}(X) \cong \widetilde{C H}^{n}(X)H~Z2n,n(X)≅CH~n(X)
For FFFFF a field, we have isomorphisms
H~Za,b(SpecF)≅{KnMW(F) for a=b=nHZa,b(SpecF) for a≠bH~Za,b(Specâ¡F)≅KnMW(F) for a=b=nHZa,b(Specâ¡F) for a≠btilde(H)Z^(a,b)(Spec F)~={[K_(n)^(MW)(F)," for "a=b=n],[HZ^(a,b)(Spec F)," for "a!=b]:}\tilde{H} \mathbb{Z}^{a, b}(\operatorname{Spec} F) \cong \begin{cases}K_{n}^{\mathrm{MW}}(F) & \text { for } a=b=n \\ H \mathbb{Z}^{a, b}(\operatorname{Spec} F) & \text { for } a \neq b\end{cases}H~Za,b(Specâ¡F)≅{KnMW(F) for a=b=nHZa,b(Specâ¡F) for a≠b
This follows from
Theorem 4.9 ([7, THEOREM 17]). Let H~Za,b,HZa,bH~Za,b,HZa,btilde(H)Z^(a,b),HZ^(a,b)\tilde{\mathscr{H}} \mathbb{Z}^{a, b}, \mathscr{H} \mathbb{Z}^{a, b}H~Za,b,HZa,b denote the respective homotopy sheaves π−a,−b(H~Z),π−a,−b(HZ)π−a,−b(H~Z),π−a,−b(HZ)pi_(-a,-b)( tilde(H)Z),pi_(-a,-b)(HZ)\pi_{-a,-b}(\tilde{H} \mathbb{Z}), \pi_{-a,-b}(H \mathbb{Z})π−a,−b(H~Z),π−a,−b(HZ). Then for a≠ba≠ba!=ba \neq ba≠b, the map
is an isomorphism. Moreover, we have canonical isomorphisms H~Zb,b=KbMWH~Zb,b=KbMWtilde(H)Z^(b,b)=K_(b)^(MW)\tilde{\mathscr{H}} \mathbb{Z}^{b, b}=\mathcal{K}_{b}^{\mathrm{MW}}H~Zb,b=KbMW, HZb,b=KbMHZb,b=KbMHZ^(b,b)=K_(b)^(M)\mathscr{H} \mathbb{Z}^{b, b}=\mathcal{K}_{b}^{M}HZb,b=KbM, and Ξa,b:H~Zb,b→Hb,bΞa,b:H~Zb,b→Hb,bXi^(a,b): tilde(H)Z^(b,b)rarrH^(b,b)\Xi^{a, b}: \tilde{\mathscr{H}} \mathbb{Z}^{b, b} \rightarrow \mathscr{H}^{b, b}Ξa,b:H~Zb,b→Hb,b is canonical surjection KbMW→KbMKbMW→KbMK_(b)^(MW)rarrK_(b)^(M)\mathcal{K}_{b}^{\mathrm{MW}} \rightarrow \mathcal{K}_{b}^{M}KbMW→KbM.
To prove Theorem 4.8, one applies this to the local-global spectral sequence
noting that HZq,n=0HZq,n=0HZ^(q,n)=0\mathscr{H} \mathbb{Z}^{q, n}=0HZq,n=0 for n<0n<0n < 0n<0n<0. This implies that the Gersten resolution of HZq,nHZq,nHZ^(q,n)\mathscr{H} \mathbb{Z}^{q, n}HZq,n has length ≤n≤n<= n\leq n≤n and thus Hp(XNis ,HZq,n)=0HpXNis ,HZq,n=0H^(p)(X_("Nis "),HZ^(q,n))=0H^{p}\left(X_{\text {Nis }}, \mathscr{H} \mathbb{Z}^{q, n}\right)=0Hp(XNis ,HZq,n)=0 for p>np>np > np>np>n.
In general, one can approximate H~Za,b(X)H~Za,b(X)tilde(H)Z^(a,b)(X)\tilde{H} \mathbb{Z}^{a, b}(X)H~Za,b(X) using the local-global sequence. Combined with Theorem 4.9 and the exact sheaf sequence
this tells us that the Milnor-Witt cohomology of XXXXX is built out of the usual motivic cohomology combined with information arising from quadratic forms.
4.4. Milnor-Witt motives
Rather than pulling the Milnor-Witt cohomology out of the motivic stable homotopy hat, there is another construction that is embedded in a Voevodsky-type triangulated category built out of a modified category of correspondences. We refer to [8] and [31] for details.
The Chow-Witt groups on a smooth XXXXX have been defined using the Rost-Schmid complex; one can also define Chow-Witt cycles with a fixed support using a modified version of the Rost-Schmid complex.
Definition 4.10. Let XXXXX be a smooth kkkkk-scheme, LLL\mathscr{L}L an invertible sheaf on XXXXX, and T⊂XT⊂XT sub XT \subset XT⊂X a closed subset. The nnnnnth LLL\mathscr{L}L-twisted Rost-Schmid complex with supports in T,RST∗(X,n;L)T,RST∗â¡(X,n;L)T,RS_(T)^(**)(X,n;L)T, \operatorname{RS}_{T}^{*}(X, n ; \mathscr{L})T,RST∗â¡(X,n;L), is the subcomplex of RS∗(X,L,n)RS∗â¡(X,L,n)RS^(**)(X,L,n)\operatorname{RS}^{*}(X, \mathscr{L}, n)RS∗â¡(X,L,n) with
The usual arguments used to prove Gersten's conjecture yield the following result.
Lemma 4.11. Let XXXXX be a smooth kkkkk-scheme, LLL\mathscr{L}L an invertible sheaf on XXXXX, and T⊂XT⊂XT sub XT \subset XT⊂X a closed subset. The cohomology with support HTp(X,KnMW(L)X)HTpX,KnMW(L)XH_(T)^(p)(X,K_(n)^(MW)(L)_(X))H_{T}^{p}\left(X, \mathcal{K}_{n}^{\mathrm{MW}}(\mathscr{L})_{X}\right)HTp(X,KnMW(L)X) is computed as
which allows us to think of HTn(X,KnMW(L)X)HTnX,KnMW(L)XH_(T)^(n)(X,K_(n)^(MW)(L)_(X))H_{T}^{n}\left(X, \mathcal{K}_{n}^{\mathrm{MW}}(\mathscr{L})_{X}\right)HTn(X,KnMW(L)X) as the group of "Grothendieck-Witt cycles" supported on TTTTT, whose definition we hinted at in the beginning of this section. We write this as Z~Tn(X,L,n)Z~Tn(X,L,n)tilde(Z)_(T)^(n)(X,L,n)\tilde{Z}_{T}^{n}(X, \mathscr{L}, n)Z~Tn(X,L,n), with the warning that this is only defined for TTTTT a closed subset of a smooth XXXXX of pure codimension nnnnn.
Note that the fact that TTTTT has pure codimension nnnnn implies that there are no relations in HTn(X,KnMW(L)X)HTnX,KnMW(L)XH_(T)^(n)(X,K_(n)^(MW)(L)_(X))H_{T}^{n}\left(X, \mathcal{K}_{n}^{\mathrm{MW}}(\mathscr{L})_{X}\right)HTn(X,KnMW(L)X) coming from K1MW(k(w))K1MW(k(w))K_(1)^(MW)(k(w))K_{1}^{\mathrm{MW}}(k(w))K1MW(k(w)) for wwwww a codimension n−1n−1n-1n-1n−1 point of XXXXX. For similar reasons, the corresponding group for the Chow groups, HTn(X,Kn,XM)HTnX,Kn,XMH_(T)^(n)(X,K_(n,X)^(M))H_{T}^{n}\left(X, \mathcal{K}_{n, X}^{M}\right)HTn(X,Kn,XM), is just the subgroup ZTn(X)ZTn(X)Z_(T)^(n)(X)Z_{T}^{n}(X)ZTn(X) of Zn(X)Zn(X)Z^(n)(X)Z^{n}(X)Zn(X) freely generated by the irreducible components of TTTTT, that is, the group of codimension nnnnn cycles on XXXXX with support contained in TTTTT.
For T⊂T′⊂XT⊂T′⊂XT subT^(')sub XT \subset T^{\prime} \subset XT⊂T′⊂X, two codimension- nnnnn closed subsets, we have the evident map Z~Tn(X,L,n)→Z~T′n(X,L,n)Z~Tn(X,L,n)→Z~T′n(X,L,n)tilde(Z)_(T)^(n)(X,L,n)rarr tilde(Z)_(T^('))^(n)(X,L,n)\tilde{Z}_{T}^{n}(X, \mathscr{L}, n) \rightarrow \tilde{Z}_{T^{\prime}}^{n}(X, \mathscr{L}, n)Z~Tn(X,L,n)→Z~T′n(X,L,n). The rank map GW(−)→ZGW(−)→ZGW(-)rarrZ\mathrm{GW}(-) \rightarrow \mathbb{Z}GW(−)→Z gives the homomorphism Z~Tn(X,L,n)→ZTn(X)Z~Tn(X,L,n)→ZTn(X)tilde(Z)_(T)^(n)(X,L,n)rarrZ_(T)^(n)(X)\tilde{Z}_{T}^{n}(X, \mathscr{L}, n) \rightarrow Z_{T}^{n}(X)Z~Tn(X,L,n)→ZTn(X).
Definition 4.12. For X,YX,YX,YX, YX,Y in SmkSmkSm_(k)\operatorname{Sm}_{k}Smk, let A(X,Y)A(X,Y)A(X,Y)\mathcal{A}(X, Y)A(X,Y) be the set of closed subsets T⊂X×YT⊂X×YT sub X xx YT \subset X \times YT⊂X×Y such that each component of TTTTT is finite over XXXXX and maps surjectively onto an irreducible component of XXXXX. We make A(X,Y)A(X,Y)A(X,Y)\mathcal{A}(X, Y)A(X,Y) a poset by the inclusion of closed subsets.
Note that if YYYYY is irreducible of dimension nnnnn, then a closed subset T⊂X×YT⊂X×YT sub X xx YT \subset X \times YT⊂X×Y is in A(X,Y)A(X,Y)A(X,Y)\mathcal{A}(X, Y)A(X,Y) if and only if TTTTT is finite over XXXXX and has pure codimension nnnnn on X×YX×YX xx YX \times YX×Y.
Definition 4.13 (Calmès-Fasel [31,§4.1][31,§4.1][31,§4.1][31, \S 4.1]§[31,§4.1] ). Let X,YX,YX,YX, YX,Y be in SmkSmkSm_(k)\operatorname{Sm}_{k}Smk and suppose YYYYY is irreducible of dimension nnnnn. Define
Corr~k(X,Y)=colimT∈A(X,Y)Z~Tn(X×Y,p2∗ωY/k)Corr~k(X,Y)=colimT∈A(X,Y)â¡Z~TnX×Y,p2∗ωY/kwidetilde(Corr)_(k)(X,Y)=colim_(T inA(X,Y)) tilde(Z)_(T)^(n)(X xx Y,p_(2)^(**)omega_(Y//k))\widetilde{\operatorname{Corr}}_{k}(X, Y)=\operatorname{colim}_{T \in \mathcal{A}(X, Y)} \tilde{Z}_{T}^{n}\left(X \times Y, p_{2}^{*} \omega_{Y / k}\right)Corr~k(X,Y)=colimT∈A(X,Y)â¡Z~Tn(X×Y,p2∗ωY/k)
Extend the definition to general YYYYY by additivity.
Using the functorial properties of pullback, intersection product and proper pushforward for the Chow-Witt groups with support, we have a well-defined composition law
The twisting by the relative dualizing sheaf in the definition of Corr~k(−,−)Corr~k(−,−)widetilde(Corr)_(k)(-,-)\widetilde{\operatorname{Corr}}_{k}(-,-)Corr~k(−,−) is exactly what is needed for the push-forward map pXZ∗pXZ∗p_(XZ)**p_{X Z} *pXZ∗ to be defined.
This defines the additive category Corr ~k Corr ~kwidetilde(" Corr ")_(k)\widetilde{\text { Corr }}_{k} Corr ~k with objects SmkSmkSm_(k)\mathrm{Sm}_{k}Smk and morphisms Corr~k(X,Y)Corr~k(X,Y)widetilde(Corr)_(k)(X,Y)\widetilde{\operatorname{Corr}}_{k}(X, Y)Corr~k(X,Y). The rank map gives an additive functor
One then follows the program used by Voevodsky to define the abelian category of Nisnevich sheaves with Milnor-Witt transfers, ShNis MW(k)ShNis MWâ¡(k)Sh_("Nis ")^(MW)(k)\operatorname{Sh}_{\text {Nis }}^{\mathrm{MW}}(k)ShNis MWâ¡(k), and then DM~eff (k)⊂DM~eff (k)⊂widetilde(DM)^("eff ")(k)sub\widetilde{\mathrm{DM}}^{\text {eff }}(k) \subsetDM~eff (k)⊂D(ShNisMWtr(k))DShNisMWtrâ¡(k)D(Sh_(Nis)^(MWtr)(k))D\left(\operatorname{Sh}_{\mathrm{Nis}}^{\mathrm{MWtr}}(k)\right)D(ShNisMWtrâ¡(k)) as the full subcategory of complexes with strictly A1A1A^(1)\mathbb{A}^{1}A1-homotopy invariant cohomology sheaves. One has the localization functor
constructed using the Suslin complex, the representable sheaves Z~tr(X)Z~tr(X)tilde(Z)^(tr)(X)\tilde{\mathbb{Z}}^{\mathrm{tr}}(X)Z~tr(X) for X∈SmkX∈SmkX inSm_(k)X \in \operatorname{Sm}_{k}X∈Smk, their corresponding motives M~eff (X):=L~A1(Z~tr (X))∈DM~eff (k)M~eff (X):=L~A1Z~tr (X)∈DM~eff (k)tilde(M)^("eff ")(X):= tilde(L)_(A^(1))( tilde(Z)^("tr ")(X))in widetilde(DM)^("eff ")(k)\tilde{M}^{\text {eff }}(X):=\tilde{L}_{\mathbb{A}^{1}}\left(\tilde{Z}^{\text {tr }}(X)\right) \in \widetilde{\mathrm{DM}}^{\text {eff }}(k)M~eff (X):=L~A1(Z~tr (X))∈DM~eff (k) and the Tate motives Z~(n)Z~(n)tilde(Z)(n)\tilde{\mathbb{Z}}(n)Z~(n) arising from the reduced motive of P1P1P^(1)\mathbb{P}^{1}P1. Finally, one constructs DM~(k)DM~(k)widetilde(DM)(k)\widetilde{\mathrm{DM}}(k)DM~(k) as a category of Z~(1)Z~(1)tilde(Z)(1)\tilde{\mathbb{Z}}(1)Z~(1) -
Hp(X,Z~(q))≅HZ~p,q(X)Hp(X,Z~(q))≅HZ~p,q(X)H^(p)(X, tilde(Z)(q))~=H tilde(Z)^(p,q)(X)H^{p}(X, \tilde{\mathbb{Z}}(q)) \cong H \tilde{\mathbb{Z}}^{p, q}(X)Hp(X,Z~(q))≅HZ~p,q(X)
The proof is very much the same as for motivic cohomology. One shows there is an equivalence of DM~(k)DM~(k)widetilde(DM)(k)\widetilde{\mathrm{DM}}(k)DM~(k) with the homotopy category of HZ~HZ~H tilde(Z)H \tilde{\mathbb{Z}}HZ~-modules (this is [8, THEOREM 5.2]). This gives an adjunction
with HZ~∧−HZ~∧−H tilde(Z)^^-H \tilde{\mathbb{Z}} \wedge-HZ~∧− the free HZ~HZ~H tilde(Z)H \tilde{\mathbb{Z}}HZ~ module functor and the Eilenberg-MacLane functor EM~EM~widetilde(EM)\widetilde{\mathrm{EM}}EM~ the forgetful functor. This gives EM~(Z~(0))=HZ~,M~(X)=HZ~∧ΣP1∞X+EM~(Z~(0))=HZ~,M~(X)=HZ~∧ΣP1∞X+widetilde(EM)( tilde(Z)(0))=H tilde(Z), tilde(M)(X)=H tilde(Z)^^Sigma_(P^(1))^(oo)X_(+)\widetilde{\operatorname{EM}}(\tilde{\mathbb{Z}}(0))=H \tilde{\mathbb{Z}}, \tilde{M}(X)=H \tilde{\mathbb{Z}} \wedge \Sigma_{\mathbb{P}^{1}}^{\infty} X_{+}EM~(Z~(0))=HZ~,M~(X)=HZ~∧ΣP1∞X+, and induces the isomorphism
5. CHOW GROUPS AND MOTIVIC COHOMOLOGY WITH MODULUS
Up to now, all the version of motivic cohomology we have considered share the A1A1A^(1)\mathbb{A}^{1}A1 homotopy invariance property, namely, that H∗(X,Z(∗))≅H∗(X×A1,Z(∗))H∗(X,Z(∗))≅H∗X×A1,Z(∗)H^(**)(X,Z(**))~=H^(**)(X xxA^(1),Z(**))H^{*}(X, \mathbb{Z}(*)) \cong H^{*}\left(X \times \mathbb{A}^{1}, \mathbb{Z}(*)\right)H∗(X,Z(∗))≅H∗(X×A1,Z(∗)); essentially by construction, this property is enjoyed by all theories that are represented in the motivic stable homotopy category. Although this is a fundamental property controlling a large collection of cohomology theories, this places a serious restriction in at least two naturally occurring areas.
One is the use of deformation theory. This relies on having useful invariants defined on non-reduced schemes, but a cohomology theory that satisfies A1A1A^(1)\mathbb{A}^{1}A1-invariance will not distinguish between a scheme and its reduced closed subscheme. The second occurs in ramification theory. An A1A1A^(1)\mathbb{A}^{1}A1-homotopy invariant theory will not detect Artin-Schreyer covers, and would not give invariants that detect wild ramification.
Fortunately, we have an interesting cohomology theory that is not A1A1A^(1)\mathbb{A}^{1}A1-homotopy invariant, namely, algebraic KKKKK-theory, that we can use as a model for a general theory. Algebraic KKKKK-theory does satisfy the A1A1A^(1)\mathbb{A}^{1}A1-invariance property when restricted to regular schemes, but in general this fails. Besides allowing KKKKK-theory to have a role in deformation theory and ramification theory, this lack of A1A1A^(1)\mathbb{A}^{1}A1-invariance gives rise to interesting invariants of singularities.
5.1. Higher Chow groups with modulus
The theory of Chow groups with modulus attempts to refine the classical theory of the Chow groups to be useful in both of these areas. This is still a theory in the process of development; just as in the early days of motivic cohomology, many approaches are inspired by properties of algebraic KKKKK-theory.
The tangent space to the functor X↦OX×X↦OX×X|->O_(X)^(xx)X \mapsto \mathcal{O}_{X}^{\times}X↦OX×is given by the structure sheaf, X↦OXX↦OXX|->O_(X)X \mapsto \mathcal{O}_{X}X↦OX, via the isomorphism
Via the isomorphism Pic(X)≅H1(X,OX×)Picâ¡(X)≅H1X,OX×Pic(X)~=H^(1)(X,O_(X)^(xx))\operatorname{Pic}(X) \cong H^{1}\left(X, \mathcal{O}_{X}^{\times}\right)Picâ¡(X)≅H1(X,OX×), this shows that the tangent space at XXXXX to the functor Pic(−)Picâ¡(−)Pic(-)\operatorname{Pic}(-)Picâ¡(−) is H1(X,OX)H1X,OXH^(1)(X,O_(X))H^{1}\left(X, \mathcal{O}_{X}\right)H1(X,OX).
In [23], Bloch computes the tangent space to K2K2K_(2)K_{2}K2 (on local QQQ\mathbb{Q}Q-algebras), giving the isomorphism of sheaves on XZarXZarX_(Zar)X_{\mathrm{Zar}}XZar (for XXXXX a QQQ\mathbb{Q}Q-scheme)
to justify defining CH2(X[ε]/(ε2))CH2X[ε]/ε2CH^(2)(X[epsi]//(epsi^(2)))\mathrm{CH}^{2}\left(X[\varepsilon] /\left(\varepsilon^{2}\right)\right)CH2(X[ε]/(ε2)) as H2(X[ε]/(ε2)Zar,K2)H2X[ε]/ε2Zar,K2H^(2)(X[epsi]//(epsi^(2))_(Zar),K_(2))H^{2}\left(X[\varepsilon] /\left(\varepsilon^{2}\right)_{\mathrm{Zar}}, \mathcal{K}_{2}\right)H2(X[ε]/(ε2)Zar,K2), giving
For XXXXX a smooth projective surface over CCC\mathbb{C}C with H2(X,OX)≠0H2X,OX≠0H^(2)(X,O_(X))!=0H^{2}\left(X, \mathcal{O}_{X}\right) \neq 0H2(X,OX)≠0, the exact sheaf sequence
⋯→ACHn(k,n−1,m+1)→ACHn(k,n−1,m)→⋯⋯→ACHn(k,n−1,m+1)→ACHn(k,n−1,m)→⋯cdots rarr ACH^(n)(k,n-1,m+1)rarr ACH^(n)(k,n-1,m)rarr cdots\cdots \rightarrow A \mathrm{CH}^{n}(k, n-1, m+1) \rightarrow A \mathrm{CH}^{n}(k, n-1, m) \rightarrow \cdots⋯→ACHn(k,n−1,m+1)→ACHn(k,n−1,m)→⋯
He showed this is endowed with additional endomorphisms FnFnF_(n)F_{n}Fn and VnVnV_(n)V_{n}Vn, and the graded group ⨁nACHn(k,n−1,∗+1)∗≥2â¨n ACHn(k,n−1,∗+1)∗≥2bigoplus_(n)ACH^(n)(k,n-1,**+1)_(** >= 2)\bigoplus_{n} A \mathrm{CH}^{n}(k, n-1, *+1)_{* \geq 2}â¨nACHn(k,n−1,∗+1)∗≥2 has the structure of a pro-differential graded algebra. In fact, we have
Theorem 5.1 (Rülling). Let kkkkk be a field of characteristic ≠2≠2!=2\neq 2≠2. The pro-dga ⨁nACHn(kâ¨n ACHn(kbigoplus_(n)ACH^(n)(k\bigoplus_{n} A \mathrm{CH}^{n}(kâ¨nACHn(k, n−1,∗+1)n−1,∗+1)n-1,**+1)n-1, *+1)n−1,∗+1), with FnFnF_(n)F_{n}Fn as Frobenius and VnVnV_(n)V_{n}Vn as Verschiebung, is isomorphic to the de RhamWitt complex of Madsen-Hesselholt,
With essentially the same definition as given by Bloch-Esnault, the additive cycle complex and additive Chow groups were extended to arbitrary kkkkk-schemes YYYYY by Park [98], replacing A1A1A^(1)\mathbb{A}^{1}A1 and divisor m⋅0mâ‹…0m*0m \cdot 0mâ‹…0 with the scheme Y×A1Y×A1Y xxA^(1)Y \times \mathbb{A}^{1}Y×A1 and divisor m⋅Y×0mâ‹…Y×0m*Y xx0m \cdot Y \times 0mâ‹…Y×0. Binda and Saito [20] went one step further, defining complexes zq(X,D,∗)zq(X,D,∗)z^(q)(X,D,**)z^{q}(X, D, *)zq(X,D,∗) for a pair (X,D)(X,D)(X,D)(X, D)(X,D) of a finite type separated kkkkk-scheme XXXXX and a Cartier divisor DDDDD, using essentially the same definition as before. The homology is the higher Chow group with modulus
The constructions of Bloch-Esnault, Park, and Binda-Saito all use a cubical model of Bloch's cycle complex. Here one replaces the algebraic nnnnn-simplex, Δkn=Speck[t0,…Δkn=Specâ¡kt0,…Delta_(k)^(n)=Spec k[t_(0),dots:}\Delta_{k}^{n}=\operatorname{Spec} k\left[t_{0}, \ldots\right.Δkn=Specâ¡k[t0,…, tn]/∑iti−1tn/∑i ti−1{:t_(n)]//sum_(i)t_(i)-1\left.t_{n}\right] / \sum_{i} t_{i}-1tn]/∑iti−1, with the algebraic nnnnn-cube
The notation means that one considers (P1∖{1})n≅AnP1∖{1}n≅An(P^(1)\\{1})^(n)~=A^(n)\left(\mathbb{P}^{1} \backslash\{1\}\right)^{n} \cong \mathbb{A}^{n}(P1∖{1})n≅An with its "faces" defined by setting some of the coordinates equal to 0 or ∞∞oo\infty∞. The corresponding cycle complex zq(X,∗)czq(X,∗)cz^(q)(X,**)_(c)z^{q}(X, *)_{c}zq(X,∗)c has degree nnnnn component zq(X,n)czq(X,n)cz^(q)(X,n)_(c)z^{q}(X, n)_{c}zq(X,n)c the codimension qqqqq cycles on X×◻nX×◻nX xxâ—»^(n)X \times \square^{n}X×◻n that intersect X×FX×FX xx FX \times FX×F properly for all faces FFFFF of ◻nâ—»nâ—»^(n)\square^{n}â—»n; one also needs to quotient out by the degenerate cycles, these being the ones that come by pullback via projection to a ◻mâ—»mâ—»^(m)\square^{m}â—»m with m<nm<nm < nm<nm<n. The differential is again an alternating sum of restrictions to the maximal faces ti=0ti=0t_(i)=0t_{i}=0ti=0 and ti=∞ti=∞t_(i)=oot_{i}=\inftyti=∞.
This complex also computes the motivic cohomology of XXXXX, just as Bloch's simplicial cycle complex does. In the Binda-Saito construction, the modulus condition arises by considering the closed box ◻¯n:=(P1)n◻¯n:=P1nbar(â—»)^(n):=(P^(1))^(n)\bar{\square}^{n}:=\left(\mathbb{P}^{1}\right)^{n}◻¯n:=(P1)n. Let Fni⊂(P1)nFni⊂P1nF_(n)^(i)sub(P^(1))^(n)F_{n}^{i} \subset\left(\mathbb{P}^{1}\right)^{n}Fni⊂(P1)n be the divisor defined by ti=1ti=1t_(i)=1t_{i}=1ti=1 and let Fn=∑i=1nFniFn=∑i=1n FniF_(n)=sum_(i=1)^(n)F_(n)^(i)F_{n}=\sum_{i=1}^{n} F_{n}^{i}Fn=∑i=1nFni. In (P1)n×XP1n×X(P^(1))^(n)xx X\left(\mathbb{P}^{1}\right)^{n} \times X(P1)n×X we have two distinguished Cartier divisors, (P1)n×DP1n×D(P^(1))^(n)xx D\left(\mathbb{P}^{1}\right)^{n} \times D(P1)n×D and Fn×XFn×XF_(n)xx XF_{n} \times XFn×X. A subvariety Z⊂(P1∖{1})n×XZ⊂P1∖{1}n×XZ sub(P^(1)\\{1})^(n)xx XZ \subset\left(\mathbb{P}^{1} \backslash\{1\}\right)^{n} \times XZ⊂(P1∖{1})n×X that is in zq(X,n)czq(X,n)cz^(q)(X,n)_(c)z^{q}(X, n)_{c}zq(X,n)c satisfies the modulus condition if
where p:Z¯N→(P1)n×Xp:Z¯N→P1n×Xp: bar(Z)^(N)rarr(P^(1))^(n)xx Xp: \bar{Z}^{N} \rightarrow\left(\mathbb{P}^{1}\right)^{n} \times Xp:Z¯N→(P1)n×X is the normalization of the closure of ZZZZZ in ◻¯n×X◻¯n×Xbar(â—»)^(n)xx X\bar{\square}^{n} \times X◻¯n×X. Restricting to the subgroup of Zq(◻n×X)Zqâ—»n×XZ^(q)(â—»^(n)xx X)Z^{q}\left(\square^{n} \times X\right)Zq(â—»n×X) generated by codimension qqqqq subvarieties Z⊂◻n×XZ⊂◻n×XZ subâ—»^(n)xx XZ \subset \square^{n} \times XZ⊂◻n×X that intersect faces properly and satisfy the modulus condition yields the cycle complex with modulus zq(X;D,∗)⊂zq(X,∗)czq(X;D,∗)⊂zq(X,∗)cz^(q)(X;D,**)subz^(q)(X,**)_(c)z^{q}(X ; D, *) \subset z^{q}(X, *)_{c}zq(X;D,∗)⊂zq(X,∗)c; the higher Chow groups with modulus is then defined as
The second construction of Bloch-Esnault, and Park's generalization, are recovered as the special cases X=Ak1X=Ak1X=A_(k)^(1)X=\mathbb{A}_{k}^{1}X=Ak1 and D=m⋅0D=mâ‹…0D=m*0D=m \cdot 0D=mâ‹…0 in the Bloch-Esnault version and X=Y×A1X=Y×A1X=Y xxA^(1)X=Y \times \mathbb{A}^{1}X=Y×A1, D=m⋅Y×0D=mâ‹…Y×0D=m*Y xx0D=m \cdot Y \times 0D=mâ‹…Y×0 in Park's version.
For XXXXX a finite type kkkkk-scheme, recall the Bloch motivic complex ZBI(q)X∗ZBI(q)X∗Z_(BI)(q)_(X)^(**)\mathbb{Z}_{\mathrm{BI}}(q)_{X}^{*}ZBI(q)X∗ defined as the Zariski sheafification of the presheaf U↦zq(X,2q−∗)U↦zq(X,2q−∗)U|->z^(q)(X,2q-**)U \mapsto z^{q}(X, 2 q-*)U↦zq(X,2q−∗) (this is already a Nisnevich sheaf). Bloch's cycle complexes satisfy an important localization property: the natural maps to Zariski and Nisnevich hypercohomology
are isomorphisms. This fails for the cycle complex with modulus, although the comparison between the Zariski and Nisnevich hypercohomology seems to be still an open question.
Iwasa and Kai consider the Nisnevich sheafification Z(q)(X;D)∗Z(q)(X;D)∗Z(q)_((X;D))^(**)\mathcal{Z}(q)_{(X ; D)}^{*}Z(q)(X;D)∗ of the presheaf
U↦zq(U;D×XU,2q−∗)U↦zqU;D×XU,2q−∗U|->z^(q)(U;Dxx_(X)U,2q-**)U \mapsto z^{q}\left(U ; D \times_{X} U, 2 q-*\right)U↦zq(U;D×XU,2q−∗)
We call Hp(XNis,Z(q)(X;D)∗)HpXNis,Z(q)(X;D)∗H^(p)(X_(Nis),Z(q)_((X;D))^(**))\mathbb{H}^{p}\left(X_{\mathrm{Nis}}, \mathcal{Z}(q)_{(X ; D)}^{*}\right)Hp(XNis,Z(q)(X;D)∗) the motivic cohomology with modulus for (X,D)(X,D)(X,D)(X, D)(X,D). Kai [74] shows that this sheafified version has contravariant functoriality. Iwasa and Kai [67] construct Chern class maps from relative KKKKK-theory
5.2. 0 -cycles with modulus and class field theory
There is a classical theory of 0 -cycles on a smooth complete curve CCCCC with a modulus condition at a finite set of points SSSSS, due to Rosenlicht and Serre [109, III]. The idea is quite simple, instead of relations coming from divisors (zeros minus poles) of an arbitrary rational function f,ff,ff,ff, ff,f is required to have a power series expansion at each point p∈Sp∈Sp in Sp \in Sp∈S, with leading term 1 and the next nonzero term of the form utpnputpnput_(p)^(n_(p))u t_{p}^{n_{p}}utpnp, with u(p)≠0,tpu(p)≠0,tpu(p)!=0,t_(p)u(p) \neq 0, t_{p}u(p)≠0,tp a local coordinate at ppppp and the integer np>0np>0n_(p) > 0n_{p}>0np>0 being the "modulus." This is applied to the class field theory of a smooth open curve U⊂CU⊂CU sub CU \subset CU⊂C over a finite field [109, THEOREM 4], that identifies the inverse limit of the groups of degree 0 cycle classes on UUUUU, with modulus supported in C∖UC∖UC\\UC \backslash UC∖U, with the kernel of the map π1et (U)ab→Gal(k¯/k)Ï€1et (U)ab→Galâ¡(k¯/k)pi_(1)^("et ")(U)^(ab)rarr Gal( bar(k)//k)\pi_{1}^{\text {et }}(U)^{a b} \rightarrow \operatorname{Gal}(\bar{k} / k)Ï€1et (U)ab→Galâ¡(k¯/k).
In their class field theory for higher-dimensional varieties, Kato and Saito [80] introduce a group of 0 -cycles on a kkkkk-scheme XXXXX with modulus DDDDD, defined by
with Kn,(X,D)MKn,(X,D)MK_(n,(X,D))^(M)\mathcal{K}_{n,(X, D)}^{M}Kn,(X,D)M a relative version of the Milnor KKKKK-theory sheaf, recalling Kato's isomorphism Hn(X,KnM)≅CHn(X)HnX,KnM≅CHn(X)H^(n)(X,K_(n)^(M))~=CH^(n)(X)H^{n}\left(X, \mathcal{K}_{n}^{M}\right) \cong \mathrm{CH}^{n}(X)Hn(X,KnM)≅CHn(X) for XXXXX a smooth kkkkk-scheme [78]. Kerz and Saito give a different definition of a group of relative 0 -cycles C(X,D)C(X,D)C(X,D)C(X, D)C(X,D) on a normal kkkkk-scheme XXXXX with effective Cartier
divisor DDDDD such that X∖DX∖DX\\DX \backslash DX∖D is smooth. It follows from their comments in [81, DEFINITION 1.6] that C(X,D)=CHn(X;D,0)C(X,D)=CHn(X;D,0)C(X,D)=CH^(n)(X;D,0)C(X, D)=\mathrm{CH}^{n}(X ; D, 0)C(X,D)=CHn(X;D,0) for XXXXX of dimension nnnnn, and it is easy to see that the KatoSaito and Kerz-Saito relative 0 -cycles agree with the Rosenlicht-Serre groups in the case of curves.
Kerz and Saito consider a smooth finite-type kkkkk-scheme UUUUU, choose a normal compactification XXXXX and define the topological group C(U):=limDC(X,D)C(U):=limD C(X,D)C(U):=lim_(D)C(X,D)C(U):=\lim _{D} C(X, D)C(U):=limDC(X,D), as DDDDD runs over effective Cartier divisors on XXXXX, supported in X∖UX∖UX\\UX \backslash UX∖U, and with each C(X,D)C(X,D)C(X,D)C(X, D)C(X,D) given the discrete topology. They show that C(U)C(U)C(U)C(U)C(U) is independent of the choice of XXXXX, and their main result generalizes class field theory for smooth curves over a finite field as described above.
There has been a great deal of interest in constructing a categorical framework for motivic cohomology with modulus. A central issue is the lack of A1A1A^(1)\mathbb{A}^{1}A1-homotopy invariance for this theory, which raises the question of what type of homotopy invariance should replace this.
One direction has been the construction of a reasonable replacement for the category of homotopy invariant Nisnevich sheaves with transfers. A non-homotopy invariant version has been developed via the theory of reciprocity sheaves, the name coming from the reciprocity laws in class field theory of curves and its relation to the group of 0 -cycles with modulus of Rosenlicht-Serre. We will say a bit about reciprocity sheaves later on, in the context of motives for log schemes Section 5.4.
For now, we will look at categories of motives with modulus constructed on the Voevodsky model by introducing a new notion of correspondence and a suitable replacement for A1A1A^(1)\mathbb{A}^{1}A1-homotopy invariance.
Looking at algebraic KKKKK-theory, the closest replacement for A1A1A^(1)\mathbb{A}^{1}A1-homotopy invariance seems to be the P1P1P^(1)\mathbb{P}^{1}P1-bundle formula
valid for a general scheme XXXXX. This has led to attempts to create a category of motives with
Here one has the problem that P1P1P^(1)\mathbb{P}^{1}P1 does not have the structure of an interval, a structure enjoyed by A1A1A^(1)\mathbb{A}^{1}A1. One considers A1A1A^(1)\mathbb{A}^{1}A1 together with "endpoints" 0,1 . Following the general theory of a site with interval, as developed by Morel-Voevodsky [94, chap. 2], one needs the multiplication map m:A1×A1→A1m:A1×A1→A1m:A^(1)xxA^(1)rarrA^(1)m: \mathbb{A}^{1} \times \mathbb{A}^{1} \rightarrow \mathbb{A}^{1}m:A1×A1→A1 to allow one to consider (A1,0,1)A1,0,1(A^(1),0,1)\left(\mathbb{A}^{1}, 0,1\right)(A1,0,1) as an abstract interval. In the construction of the cycle complex with modulus, one identifies (A1,0,1)A1,0,1(A^(1),0,1)\left(\mathbb{A}^{1}, 0,1\right)(A1,0,1) with (P1∖{1},0,∞)P1∖{1},0,∞(P^(1)\\{1},0,oo)\left(\mathbb{P}^{1} \backslash\{1\}, 0, \infty\right)(P1∖{1},0,∞), and the corresponding multiplication map m′:(P1∖{1})×(P1∖{1})→m′:P1∖{1}×P1∖{1}→m^('):(P^(1)\\{1})xx(P^(1)\\{1})rarrm^{\prime}:\left(\mathbb{P}^{1} \backslash\{1\}\right) \times\left(\mathbb{P}^{1} \backslash\{1\}\right) \rightarrowm′:(P1∖{1})×(P1∖{1})→P1∖{1}P1∖{1}P^(1)\\{1}\mathbb{P}^{1} \backslash\{1\}P1∖{1} only extends as a rational map P1×P1→P1P1×P1→P1P^(1)xxP^(1)rarrP^(1)\mathbb{P}^{1} \times \mathbb{P}^{1} \rightarrow \mathbb{P}^{1}P1×P1→P1. However, m′m′m^(')m^{\prime}m′ becomes a morphism
after blowing up the point (1,1)(1,1)(1,1)(1,1)(1,1), which suggests that one should consider the closure of the graph of m′m′m^(')m^{\prime}m′ in P1×P1×P1P1×P1×P1P^(1)xxP^(1)xxP^(1)\mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{1}P1×P1×P1 as an allowable correspondence from P1×P1P1×P1P^(1)xxP^(1)\mathbb{P}^{1} \times \mathbb{P}^{1}P1×P1 to P1P1P^(1)\mathbb{P}^{1}P1.
With this as starting point, Kahn, Miyazaki, Saito, and Yamazaki [69-71] follow Voevodsky's program, defining a category of modulus correspondences MCor kk_(k)_{k}k. Objects are pairs ( M¯,M∞M¯,M∞bar(M),M^(oo)\bar{M}, M^{\infty}M¯,M∞ ) with M¯M¯bar(M)\bar{M}M¯ a separated finite-type kkkkk-scheme and M∞M∞M^(oo)M^{\infty}M∞ an effective Cartier divisor on M¯M¯bar(M)\bar{M}M¯ such that the open complement M∘:=M¯∖M∞M∘:=M¯∖M∞M^(@):= bar(M)\\M^(oo)M^{\circ}:=\bar{M} \backslash M^{\infty}M∘:=M¯∖M∞ is smooth. The morphism group
ZZZZZ (finite and surjective over a component of M∘M∘M^(@)M^{\circ}M∘ ) such that
(i) The closure Z¯Z¯bar(Z)\bar{Z}Z¯ of ZZZZZ in M¯×N¯M¯×N¯bar(M)xx bar(N)\bar{M} \times \bar{N}M¯×N¯ is proper over M¯M¯bar(M)\bar{M}M¯ (not necessarily finite).
(ii) Let f:Z¯N→M¯×N¯f:Z¯N→M¯×N¯f: bar(Z)^(N)rarr bar(M)xx bar(N)f: \bar{Z}^{N} \rightarrow \bar{M} \times \bar{N}f:Z¯N→M¯×N¯ be the normalization of Z¯Z¯bar(Z)\bar{Z}Z¯. Then
The composition law in CorkCorkCor_(k)\mathrm{Cor}_{k}Cork preserves conditions (i) and (ii), giving the category MCor_kMCor_kMCor__(k)\underline{M C o r}_{k}MCor_k with functor M_Cork→CorkM_Cork→CorkM_Cor_(k)rarrCor_(k)\underline{\mathrm{M}} \operatorname{Cor}_{k} \rightarrow \operatorname{Cor}_{k}M_Cork→Cork sending (M¯,M∞)M¯,M∞(( bar(M)),M^(oo))\left(\bar{M}, M^{\infty}\right)(M¯,M∞) to M∘M∘M^(@)M^{\circ}M∘ and with M_Cork((M¯,M∞)M_Corkâ¡M¯,M∞M_Cor_(k)((( bar(M)),M^(oo)):}\underline{\operatorname{M}} \operatorname{Cor}_{k}\left(\left(\bar{M}, M^{\infty}\right)\right.M_Corkâ¡((M¯,M∞), (N¯,N∞))→Cork(M∘,N∘)N¯,N∞→Corkâ¡M∘,N∘{:(( bar(N)),N^(oo)))rarrCor_(k)(M^(@),N^(@))\left.\left(\bar{N}, N^{\infty}\right)\right) \rightarrow \operatorname{Cor}_{k}\left(M^{\circ}, N^{\circ}\right)(N¯,N∞))→Corkâ¡(M∘,N∘) the inclusion. The product of pairs makes MCork_MCork_MCor_(k)_\underline{\mathrm{MCor}_{k}}MCork_ a symmetric monoidal category and the functor to CorkCorkCor_(k)\mathrm{Cor}_{k}Cork is symmetric monoidal.
Let ◻¯â—»Â¯bar(â—»)\bar{\square}◻¯ be the object (P1,{1})P1,{1}(P^(1),{1})\left(\mathbb{P}^{1},\{1\}\right)(P1,{1}). As hinted above, the closure of the graph of m′:(P1∖{1})×(P1∖{1})→P1∖{1}m′:P1∖{1}×P1∖{1}→P1∖{1}m^('):(P^(1)\\{1})xx(P^(1)\\{1})rarrP^(1)\\{1}m^{\prime}:\left(\mathbb{P}^{1} \backslash\{1\}\right) \times\left(\mathbb{P}^{1} \backslash\{1\}\right) \rightarrow \mathbb{P}^{1} \backslash\{1\}m′:(P1∖{1})×(P1∖{1})→P1∖{1} defines a morphism m:◻¯×◻¯→◻¯m:◻¯×◻¯→◻¯m: bar(â—»)xx bar(â—»)rarr bar(â—»)m: \bar{\square} \times \bar{\square} \rightarrow \bar{\square}m:◻¯×◻¯→◻¯ in MCor kk_(k)_{k}k.
They then consider the abelian category of additive presheaves of abelian groups
pairs (X,D)(X,D)(X,D)(X, D)(X,D), with XXXXX a proper kkkkk-scheme, as a full subcategory of MCor _k MCor _k" MCor "__(k)\underline{\text { MCor }}{ }_{k} MCor _k, with its presheaf category MPSTkMPSTkMPST_(k)\operatorname{MPST}_{k}MPSTk.
They define a category of effective proper motives with modulus, MDMeff(k)MDMeffâ¡(k)MDM^(eff)(k)\operatorname{MDM}^{\mathrm{eff}}(k)MDMeffâ¡(k), by localizing the derived category D(MPSTk)DMPSTkD(MPST_(k))D\left(\operatorname{MPST}_{k}\right)D(MPSTk). Roughly speaking, they follow the Voevodsky program, replacing the A1A1A^(1)\mathbb{A}^{1}A1-localization with ◻¯â—»Â¯bar(â—»)\bar{\square}◻¯ localization. To get the proper Nisnevich localization is a bit technical; we refer the reader to [71, DEFINITION 1.3.9] for details.
There is still quite a bit that is not known. One central problem is how to realize the various constructions of the higher Chow groups with modulus as morphisms in a suitable triangulated category. There is a connection, at least for the modulus version of Suslin homology and the Suslin complex, which we now describe.
One can show that the cubical version of the Suslin complex
is naturally quasi-isomorphic to the simplicial version C∗Sus (X)(Y)C∗Sus (X)(Y)C_(**)^("Sus ")(X)(Y)C_{*}^{\text {Sus }}(X)(Y)C∗Sus (X)(Y), where / degn means taking the quotient by the image of the pullback maps via the projections Y×◻n→Y×Y×◻n→Y×Y xxâ—»^(n)rarr Y xxY \times \square^{n} \rightarrow Y \timesY×◻n→Y×◻n−1â—»n−1â—»^(n-1)\square^{n-1}â—»n−1. For a modulus pair (X,D)(X,D)(X,D)(X, D)(X,D), one can similarly form the naive Suslin complex
Next, there is a derived Suslin complex RC∗Sus (X,D)c(−)RC∗Sus (X,D)c(−)RC_(**)^("Sus ")(X,D)_(c)(-)R C_{*}^{\text {Sus }}(X, D)_{c}(-)RC∗Sus (X,D)c(−) with a natural map of presheaves
C∗Sus(X,D)c(−)→RC∗Sus(X,D)c(−)C∗Sus(X,D)c(−)→RC∗Sus(X,D)c(−)C_(**)^(Sus)(X,D)_(c)(-)rarr RC_(**)^(Sus)(X,D)_(c)(-)C_{*}^{\mathrm{Sus}}(X, D)_{c}(-) \rightarrow R C_{*}^{\mathrm{Sus}}(X, D)_{c}(-)C∗Sus(X,D)c(−)→RC∗Sus(X,D)c(−)
By [71, THEOREM 2], for (X,D)(X,D)(X,D)(X, D)(X,D) a proper modulus pair, RC∗Sus (X,D)c(−)RC∗Sus (X,D)c(−)RC_(**)^("Sus ")(X,D)_(c)(-)R C_{*}^{\text {Sus }}(X, D)_{c}(-)RC∗Sus (X,D)c(−) computes the maps in MDMeff(k)MDMeffâ¡(k)MDM^(eff)(k)\operatorname{MDM}^{\mathrm{eff}}(k)MDMeffâ¡(k) as
However, one should not expect that the Suslin complex or its derived version should yield a version of the higher Chow groups. If one looks back at the setting of DM(k)DM(k)DM(k)\mathrm{DM}(k)DM(k), the object that most naturally yields the higher Chow groups for an arbitrary finite type kkkkk-scheme XXXXX is the motive with compact supports Mc(X)Mc(X)M^(c)(X)M^{c}(X)Mc(X). This is defined as C∗Sus (Ztrc(X))C∗Sus Ztrc(X)C_(**)^("Sus ")(Z_(tr)^(c)(X))C_{*}^{\text {Sus }}\left(\mathbb{Z}_{\mathrm{tr}}^{c}(X)\right)C∗Sus (Ztrc(X)), where Ztrc(X)Ztrc(X)Z_(tr)^(c)(X)\mathbb{Z}_{\mathrm{tr}}^{c}(X)Ztrc(X) is the presheaf with transfers with Ztrc(X)(Y)Ztrc(X)(Y)Z_(tr)^(c)(X)(Y)\mathbb{Z}_{\mathrm{tr}}^{c}(X)(Y)Ztrc(X)(Y) the free abelian group on integral W⊂Y×XW⊂Y×XW sub Y xx XW \subset Y \times XW⊂Y×X, with W→YW→YW rarr YW \rightarrow YW→Y quasi-finite and dominant over a component of Y∈SmkY∈SmkY inSm_(k)Y \in \operatorname{Sm}_{k}Y∈Smk. See [127, cHAP. 5, PROPOSITION 4.2.9] for the relation of Mc(X)Mc(X)M^(c)(X)M^{c}(X)Mc(X) with Bloch's higher Chow groups.
One can define a similar version with modulus as the object Mc(X,D)Mc(X,D)M^(c)(X,D)M^{c}(X, D)Mc(X,D) associated to the presheaf Ztrc(X,D)Ztrc(X,D)Z_(tr)^(c)(X,D)\mathbb{Z}_{\mathrm{tr}}^{c}(X, D)Ztrc(X,D), with Ztrc(X,D)(Y,E)⊂ZdimY(Y×X)Ztrc(X,D)(Y,E)⊂Zdimâ¡Y(Y×X)Z_(tr)^(c)(X,D)(Y,E)subZ_(dim Y)(Y xx X)\mathbb{Z}_{\mathrm{tr}}^{c}(X, D)(Y, E) \subset Z_{\operatorname{dim} Y}(Y \times X)Ztrc(X,D)(Y,E)⊂Zdimâ¡Y(Y×X) the subgroup generated by closed subvarieties W⊂(Y∖E)×(X∖D)W⊂(Y∖E)×(X∖D)W sub(Y\\E)xx(X\\D)W \subset(Y \backslash E) \times(X \backslash D)W⊂(Y∖E)×(X∖D) that are quasi-finite and dominant over YYYYY, and with the usual modulus condition, that the normalization ν:W¯N→ν:W¯N→nu: bar(W)^(N)rarr\nu: \bar{W}^{N} \rightarrowν:W¯N→Y×XY×XY xx XY \times XY×X of the closure of WWWWW in Y×XY×XY xx XY \times XY×X satisfies
v∗(E×X)≥v∗(Y×D)v∗(E×X)≥v∗(Y×D)v^(**)(E xx X) >= v^(**)(Y xx D)v^{*}(E \times X) \geq v^{*}(Y \times D)v∗(E×X)≥v∗(Y×D)
There is an analog of Suslin's comparison theorem in the affine case, due to KaiMiyazaki [75]: They define an equi-dimensional cycle complex with modulus
which for d=0d=0d=0d=0d=0 is the Suslin complex with modulus C∗Sus (Ztrc(X,D))(Speck,∅)C∗Sus Ztrc(X,D)(Specâ¡k,∅)C_(**)^("Sus ")(Z_(tr)^(c)(X,D))(Spec k,O/)C_{*}^{\text {Sus }}\left(\mathbb{Z}_{\mathrm{tr}}^{c}(X, D)\right)(\operatorname{Spec} k, \emptyset)C∗Sus (Ztrc(X,D))(Specâ¡k,∅)
Theorem 5.3 (Kai-Miyazaki). Let (X,D)(X,D)(X,D)(X, D)(X,D) be a modulus pair, with XXXXX affine. Then there is a pro-isomorphism
{H∗(zdequi (X,mD,∗))}m≅{CHd(X,mD,∗)}mH∗zdequi (X,mD,∗)m≅CHd(X,mD,∗)m{H_(**)(z_(d)^("equi ")(X,mD,**))}_(m)~={CH_(d)(X,mD,**)}_(m)\left\{H_{*}\left(z_{d}^{\text {equi }}(X, m D, *)\right)\right\}_{m} \cong\left\{\mathrm{CH}_{d}(X, m D, *)\right\}_{m}{H∗(zdequi (X,mD,∗))}m≅{CHd(X,mD,∗)}m
Miyazaki [91] has defined objects zequi (X,D,d)∈MNSTkzequi (X,D,d)∈MNSTkz^("equi ")(X,D,d)inMNST_(k)z^{\text {equi }}(X, D, d) \in \operatorname{MNST}_{k}zequi (X,D,d)∈MNSTk, with Ztr c(X,D)=Ztr c(X,D)=Z_("tr ")^(c)(X,D)=\mathbb{Z}_{\text {tr }}^{c}(X, D)=Ztr c(X,D)=zequi (X,D,0)zequi (X,D,0)z^("equi ")(X,D,0)z^{\text {equi }}(X, D, 0)zequi (X,D,0). The sheaf zequi (X,D,r)zequi (X,D,r)z^("equi ")(X,D,r)z^{\text {equi }}(X, D, r)zequi (X,D,r) is defined similarly to Ztrc(X,D)Ztrc(X,D)Z_(tr)^(c)(X,D)\mathbb{Z}_{\mathrm{tr}}^{c}(X, D)Ztrc(X,D), with zequi (X,Dzequi (X,Dz^("equi ")(X,Dz^{\text {equi }}(X, Dzequi (X,D, d)(Y,E)d)(Y,E)d)(Y,E)d)(Y, E)d)(Y,E) the group of cycles on (Y∖E)×(X∖D)(Y∖E)×(X∖D)(Y\\E)xx(X\\D)(Y \backslash E) \times(X \backslash D)(Y∖E)×(X∖D) generated by closed, integral W⊂(Y∖E)×(X∖D)W⊂(Y∖E)×(X∖D)W sub(Y\\E)xx(X\\D)W \subset(Y \backslash E) \times(X \backslash D)W⊂(Y∖E)×(X∖D) that are equi-dimensional of dimension ddddd over Y∖EY∖EY\\EY \backslash EY∖E, dominate a component of Y∖EY∖EY\\EY \backslash EY∖E, and with v:W¯N→Y×Xv:W¯N→Y×Xv: bar(W)^(N)rarr Y xx Xv: \bar{W}^{N} \rightarrow Y \times Xv:W¯N→Y×X satisfying the modulus condition
v∗(E×X)≥v∗(Y×D)v∗(E×X)≥v∗(Y×D)v^(**)(E xx X) >= v^(**)(Y xx D)v^{*}(E \times X) \geq v^{*}(Y \times D)v∗(E×X)≥v∗(Y×D)
Moreover, for an arbitrary modulus pair (X,D)(X,D)(X,D)(X, D)(X,D), one has
For a proper modulus pair, let Mc(X,D)Mc(X,D)M^(c)(X,D)M^{c}(X, D)Mc(X,D) denote the image of Ztr c(X,D)Ztr c(X,D)Z_("tr ")^(c)(X,D)\mathbb{Z}_{\text {tr }}^{c}(X, D)Ztr c(X,D) in MDMeff(k)MDMeffâ¡(k)MDM^(eff)(k)\operatorname{MDM}^{\mathrm{eff}}(k)MDMeffâ¡(k). One can ask if there are analogs of the theorem of Kahn-Miyazaki-Saito-Yamazaki.
Question 5.4. For (X,D)(X,D)(X,D)(X, D)(X,D) a proper modulus pair, are the maps
This uses the duality M(X)c≅M(X)∨(d)[2d]M(X)c≅M(X)∨(d)[2d]M(X)^(c)~=M(X)^(vv)(d)[2d]M(X)^{c} \cong M(X)^{\vee}(d)[2 d]M(X)c≅M(X)∨(d)[2d] for XXXXX of dimension ddddd (valid in characteristic zero, or after inverting ppppp in characteristic p>0p>0p > 0p>0p>0 ), and the extension of Suslin's quasi-isomorphism zqequi (X,∗)↪zq(X,∗)zqequi (X,∗)↪zq(X,∗)z_(q)^("equi ")(X,**)↪z_(q)(X,**)z_{q}^{\text {equi }}(X, *) \hookrightarrow z_{q}(X, *)zqequi (X,∗)↪zq(X,∗) to arbitrary XXXXX. Moreover, we have M(X)c=M(X)c=M(X)^(c)=M(X)^{c}=M(X)c=M(X)M(X)M(X)M(X)M(X) for XXXXX smooth and proper.
However, a corresponding motivic cohomology of modulus pairs seems to need a larger category. This is hinted at by the use of the duality (in DM(k))M(X)c≅DMâ¡(k))M(X)c≅DM(k))M(X)^(c)~=\operatorname{DM}(k)) M(X)^{c} \congDMâ¡(k))M(X)c≅M(X)∨(d)[2d]M(X)∨(d)[2d]M(X)^(vv)(d)[2d]M(X)^{\vee}(d)[2 d]M(X)∨(d)[2d] in the computations described above. This says in particular that each motive M(X)M(X)M(X)M(X)M(X) admits a "twisted" dual in DMeff(k)DMeff(k)DM^(eff)(k)\mathrm{DM}^{\mathrm{eff}}(k)DMeff(k), more precisely, the usual evaluation
and coevaluation maps associated with a dual exist, but as maps with target or source some Z(d)[2d]Z(d)[2d]Z(d)[2d]\mathbb{Z}(d)[2 d]Z(d)[2d] rather than the unit. For a general proper modulus pair (X,D)(X,D)(X,D)(X, D)(X,D), this does not seem to be the case; one seems to need modulus pairs with an anti-effective Cartier divisor. Another way to say the same thing, if one looks for a proper modulus pair (X,D′)X,D′(X,D^('))\left(X, D^{\prime}\right)(X,D′) such that HomMDMeff (k)(M(X,D′),Z(q)[p])HomMDMeff â¡(k)â¡MX,D′,Z(q)[p]Hom_(MDM^("eff ")(k))(M(X,D^(')),Z(q)[p])\operatorname{Hom}_{\operatorname{MDM}^{\text {eff }}(k)}\left(M\left(X, D^{\prime}\right), \mathbb{Z}(q)[p]\right)HomMDMeff â¡(k)â¡(M(X,D′),Z(q)[p]) looks at all like CHq(X,D,2q−p)CHq(X,D,2q−p)CH^(q)(X,D,2q-p)\mathrm{CH}^{q}(X, D, 2 q-p)CHq(X,D,2q−p) for some given proper modulus pair (X,D)(X,D)(X,D)(X, D)(X,D), the defining inequalities in Corr kk_(k){ }_{k}k suggest that D′D′D^(')D^{\prime}D′ could be −D−D-D-D−D. See the section "Perspectives" in [71, InTRoduction] for further details in this direction.
5.4. Logarithmic motives and reciprocity sheaves
Grothendieck motives for log schemes have been constructed in [66], where a version for mixed motives has also been constructed using systems of realizations. There the emphasis is on versions of motives for homological or numerical equivalence in the setting of logloglog\loglog schemes. In this section we discuss a recent construction of a triangulated category of log motives, by Binda-Park-Østvær [19], that follows the Voevodsky program. We refer the reader to the lectures notes of Ogus [97] for the facts about log schemes.
Recall that a logloglog\loglog scheme is a pair (X,α:M→(OX,×))X,α:M→OX,×(X,alpha:Mrarr(O_(X),xx))\left(X, \alpha: \mathcal{M} \rightarrow\left(\mathcal{O}_{X}, \times\right)\right)(X,α:M→(OX,×)) consisting of a scheme XXXXX and a homomorphism of sheaves of commutative monoids α:M→(OX,×)α:M→OX,×alpha:Mrarr(O_(X),xx)\alpha: \mathcal{M} \rightarrow\left(\mathcal{O}_{X}, \times\right)α:M→(OX,×) such that α−1(OX×)→OX×α−1OX×→OX×alpha^(-1)(O_(X)^(xx))rarrO_(X)^(xx)\alpha^{-1}\left(\mathcal{O}_{X}^{\times}\right) \rightarrow \mathcal{O}_{X}^{\times}α−1(OX×)→OX×is an isomorphism; without this last condition, the pair (X,α:M→(X,α:M→(X,alpha:Mrarr(X, \alpha: \mathcal{M} \rightarrow(X,α:M→(OX,×))OX,×{:(O_(X),xx))\left.\left(\mathcal{O}_{X}, \times\right)\right)(OX,×)) is called a pre-log structure. A pre-log structure α:M→(OX,×)α:M→OX,×alpha:Mrarr(O_(X),xx)\alpha: \mathcal{M} \rightarrow\left(\mathcal{O}_{X}, \times\right)α:M→(OX,×) induces a log structure αlog:Mlog→(OX,×)αlog:Mlog→OX,×alpha^(log):M^(log)rarr(O_(X),xx)\alpha^{\log }: \mathcal{M}^{\log } \rightarrow\left(\mathcal{O}_{X}, \times\right)αlog:Mlog→(OX,×) by taking MlogMlogM^(log)\mathcal{M}^{\log }Mlog to be the push-out (in the category of sheaves of monoids) in
Replacing the category of smooth kkkkk-schemes is the category 1Smk1Smk1Sm_(k)1 \mathrm{Sm}_{k}1Smk of fine, saturated, log smooth and separated log schemes over the log scheme Spec kkkkk endowed with the trivial logloglog\loglog structure. We refer the reader to [19] for details; one needs these technical conditions to construct the category of finite log correspondences. We call a separated, fine, saturated log scheme an fs log scheme.
We sketch the construction of the category of finite log correspondences, and describe how Binda-Park- stvær follow Voevodsky's program to define the triangulated category logDMeff (k)logâ¡DMeff (k)log DM^("eff ")(k)\log \mathrm{DM}^{\text {eff }}(k)logâ¡DMeff (k) of effective log motives over kkkkk.
For X∈SmmkX∈SmmkX inSmm_(k)X \in \operatorname{Smm}_{k}X∈Smmk, let X_X_X_\underline{X}X_ denote the underlying kkkkk-scheme. We let X∘⊂X_X∘⊂X_X^(@)subX_X^{\circ} \subset \underline{X}X∘⊂X_ denote the maximal open subscheme over which the log structure MX→OXMX→OXM_(X)rarrO_(X)\mathcal{M}_{X} \rightarrow \mathcal{O}_{X}MX→OX is trivial, that is, MX∣U=OU×MX∣U=OU×M_(X∣U)=O_(U)^(xx)\mathcal{M}_{X \mid U}=\mathcal{O}_{U}^{\times}MX∣U=OU×, and let ∂X=X_∖X∘∂X=X_∖X∘del X=X_\\X^(@)\partial X=\underline{X} \backslash X^{\circ}∂X=X_∖X∘.
Definition 5.5. 1. For X,Y∈SmkX,Y∈SmkX,Y inSm_(k)X, Y \in \operatorname{Sm}_{k}X,Y∈Smk, the group lCork(X,Y)lCorkâ¡(X,Y)lCor_(k)(X,Y)\operatorname{lCor}_{k}(X, Y)lCorkâ¡(X,Y) consisting of finite log correspondences from XXXXX to YYYYY is the free abelian group on integral closed subschemes Z_⊂X_×Y_Z_⊂X_×Y_Z_subX_xxY_\underline{Z} \subset \underline{X} \times \underline{Y}Z_⊂X_×Y_ such that
(i) Z_→X_Z_→X_Z_rarrX_\underline{Z} \rightarrow \underline{X}Z_→X_ is finite and is surjective to a component of X_X_X_\underline{X}X_.
(ii) Let ZNZNZ^(N)Z^{N}ZN be the log scheme with underlying scheme the normalization ν:Z_N→ν:Z_N→nu:Z_^(N)rarr\nu: \underline{Z}^{N} \rightarrowν:Z_N→X×YX×YX xx YX \times YX×Y of Z_Z_Z_\underline{Z}Z_ and logloglog\loglog structure (ν∘p1)log∗MX→OZNν∘p1log∗MX→OZN(nu@p_(1))_(log)^(**)M_(X)rarrO_(Z^(N))\left(\nu \circ p_{1}\right)_{\log }^{*} \mathcal{M}_{X} \rightarrow \mathcal{O}_{Z^{N}}(ν∘p1)log∗MX→OZN. Here MX→OXMX→OXM_(X)rarrO_(X)\mathcal{M}_{X} \rightarrow \mathcal{O}_{X}MX→OX is the given log structure on XXXXX and (ν∘p1)log∗MX→OZν∘p1log∗MX→OZ(nu@p_(1))_(log)^(**)M_(X)rarrO_(Z)\left(\nu \circ p_{1}\right)_{\log }^{*} \mathcal{M}_{X} \rightarrow \mathcal{O}_{Z}(ν∘p1)log∗MX→OZ is the log structure induced by the pre-log structure (ν∘p1)−1MX→(ν∘p1)−1OX→OZν∘p1−1MX→ν∘p1−1OX→OZ(nu@p_(1))^(-1)M_(X)rarr(nu@p_(1))^(-1)O_(X)rarrO_(Z)\left(\nu \circ p_{1}\right)^{-1} \mathcal{M}_{X} \rightarrow\left(\nu \circ p_{1}\right)^{-1} \mathcal{O}_{X} \rightarrow \mathcal{O}_{Z}(ν∘p1)−1MX→(ν∘p1)−1OX→OZ. Then the map of schemes p2∘ν:Z_N→Y_p2∘ν:Z_N→Y_p_(2)@nu:Z_^(N)rarrY_p_{2} \circ \nu: \underline{Z}^{N} \rightarrow \underline{Y}p2∘ν:Z_N→Y_ extends to a map of logloglog\loglog schemes ZN→YZN→YZ^(N)rarr YZ^{N} \rightarrow YZN→Y.
Remark 5.6. It follows from (i) and (ii) above that, for Z_∈lork(X,Y)Z_∈lorkâ¡(X,Y)Z_inlor_(k)(X,Y)\underline{Z} \in \operatorname{lor}_{k}(X, Y)Z_∈lorkâ¡(X,Y), the restriction of Z_Z_Z_\underline{Z}Z_ to a cycle on the open subset X∘×Y∘X∘×Y∘X^(@)xxY^(@)X^{\circ} \times Y^{\circ}X∘×Y∘ of X_×Y_X_×Y_X_xxY_\underline{X} \times \underline{Y}X_×Y_ actually lands in Cork(X∘,Y∘)Corkâ¡X∘,Y∘Cor_(k)(X^(@),Y^(@))\operatorname{Cor}_{k}\left(X^{\circ}, Y^{\circ}\right)Corkâ¡(X∘,Y∘). Moreover, by [19, LemMA 2.3.1], if the extension in (ii) exists, it is unique, so there is no need to include this as part of the data. In particular, the restriction map lork(X,Y)→Cork(X∘,Y∘)lorkâ¡(X,Y)→Corkâ¡X∘,Y∘lor_(k)(X,Y)rarrCor_(k)(X^(@),Y^(@))\operatorname{lor}_{k}(X, Y) \rightarrow \operatorname{Cor}_{k}\left(X^{\circ}, Y^{\circ}\right)lorkâ¡(X,Y)→Corkâ¡(X∘,Y∘) is injective ([19, LEMMA 2.3.2]).
The condition that there exists a map of logloglog\loglog schemes (ZN,(p1∘ν)log∗MX)→ZN,p1∘νlog∗MX→(Z^(N),(p_(1)@nu)_(log)^(**)M_(X))rarr\left(Z^{N},\left(p_{1} \circ \nu\right)_{\log }^{*} \mathcal{M}_{X}\right) \rightarrow(ZN,(p1∘ν)log∗MX)→(Y,MY)Y,MY(Y,M_(Y))\left(Y, \mathcal{M}_{Y}\right)(Y,MY) extending p2∘v:Z_N→Y_p2∘v:Z_N→Y_p_(2)@v:Z_^(N)rarrY_p_{2} \circ v: \underline{Z}^{N} \rightarrow \underline{Y}p2∘v:Z_N→Y_ is analogous to the modulus condition
v∗(D×Y)≥v∗(X×E)v∗(D×Y)≥v∗(X×E)v^(**)(D xx Y) >= v^(**)(X xx E)v^{*}(D \times Y) \geq v^{*}(X \times E)v∗(D×Y)≥v∗(X×E)
for a subvariety W⊂X∖D×Y∖EW⊂X∖D×Y∖EW sub X\\D xx Y\\EW \subset X \backslash D \times Y \backslash EW⊂X∖D×Y∖E to define a finite correspondence of modulus pairs from (X,D)(X,D)(X,D)(X, D)(X,D) to (Y,E)(Y,E)(Y,E)(Y, E)(Y,E).
that is compatible with the composition law in CorkCorkCor_(k)\mathrm{Cor}_{k}Cork via the respective restriction maps.
This defines the additive category of finite log correspondences 1Cork1Cork1Cor_(k)\mathrm{1Cor}_{k}1Cork with the same objects as for 1Smk1Smk1Sm_(k)1 \mathrm{Sm}_{k}1Smk, giving the category of presheaves with log transfers, PSSTkPSSTkPSST_(k)\mathrm{PSST}_{k}PSSTk, defined as the category of additive presheaves of abelian groups on 1lork1lork1lor_(k)1 \operatorname{lor}_{k}1lork. For a log scheme X∈1SmkX∈1SmkX in1Sm_(k)X \in 1 \mathrm{Sm}_{k}X∈1Smk, let Zltr(X)Zltr(X)Z_(ltr)(X)\mathbb{Z}_{\mathrm{ltr}}(X)Zltr(X) denote the representable presheaf
The fiber product of log schemes induces a tensor product structure on lPSTklPSTklPST_(k)\operatorname{lPST}_{k}lPSTk.
The next step is to define the log version of the Nisnevich topology.
A morphism of log schemes f:(X,MX→OX),(Y,MY→OY)f:X,MX→OX,Y,MY→OYf:(X,M_(X)rarrO_(X)),(Y,M_(Y)rarrO_(Y))f:\left(X, \mathcal{M}_{X} \rightarrow \mathcal{O}_{X}\right),\left(Y, \mathcal{M}_{Y} \rightarrow \mathcal{O}_{Y}\right)f:(X,MX→OX),(Y,MY→OY) is strict if the map of logloglog\loglog structures f∗MY→MXf∗MY→MXf^(**)M_(Y)rarrM_(X)f^{*} \mathcal{M}_{Y} \rightarrow \mathcal{M}_{X}f∗MY→MX is an isomorphism. An elementary log Nisnevich square is a cartesian square in the category of fs log schemes
A log modification is a generalization of the notion of a log blow-up, which in turn is a morphism of log schemes modeled on the birational morphism of toric varieties given by a subdivision of the fan defining the target. We refer the reader to [19, APPENDIX A] for details. The Grothendieck topology generated by the log modifications and strict Nisnevich elementary squares is called the dividing Nisnevich topology on fs log schemes. In a sense, this is a log version of the cdh topology, where all the modifications are taking place in the boundary.
With this topology in hand, we have the subcategory lNSTklNSTklNST_(k)\operatorname{lNST}_{k}lNSTk of IPSTkIPSTkIPST_(k)\operatorname{IPST}_{k}IPSTk of Nisnevich sheaves with log transfers, just as for NSTk⊂PSTkNSTk⊂PSTkNST_(k)subPST_(k)\mathrm{NST}_{k} \subset \mathrm{PST}_{k}NSTk⊂PSTk, by requiring that a presheaf with log transfers be a sheaf for the dividing Nisnevich topology when restricted to SmkSmkSm_(k)\mathrm{Sm}_{k}Smk.
Finally, we need a suitable interval object to define a good notion of homotopy invariance. This is just as for the category MDMeff (k)MDMeff â¡(k)MDM^("eff ")(k)\operatorname{MDM}^{\text {eff }}(k)MDMeff â¡(k), where we consider ◻¯â—»Â¯bar(â—»)\bar{\square}◻¯ as the scheme P1P1P^(1)\mathbb{P}^{1}P1 with compactifying log structure for (P1,{1})P1,{1}(P^(1),{1})\left(\mathbb{P}^{1},\{1\}\right)(P1,{1}). The product log scheme ◻¯2◻¯2bar(â—»)^(2)\bar{\square}^{2}◻¯2 also has the compactifying log structure for the divisor 1×P1+P1×11×P1+P1×11xxP^(1)+P^(1)xx11 \times \mathbb{P}^{1}+\mathbb{P}^{1} \times 11×P1+P1×1. However, the closure Γ¯mΓ¯mbar(Gamma)_(m)\bar{\Gamma}_{m}Γ¯m of the graph of the multiplication map m:◻¯2→◻¯m:◻¯2→◻¯m: bar(â—»)^(2)rarr bar(â—»)m: \bar{\square}^{2} \rightarrow \bar{\square}m:◻¯2→◻¯ is not a morphism m~m~tilde(m)\tilde{m}m~ in 1Cork1Cork1Cor_(k)1 \operatorname{Cor}_{k}1Cork, as the requirement that the map of Γ¯mΓ¯mbar(Gamma)_(m)\bar{\Gamma}_{m}Γ¯m to ◻¯2◻¯2bar(â—»)^(2)\bar{\square}^{2}◻¯2 be finite is not satisfied.
Another way to look at this is to note that the projection Γ¯m→◻¯2Γ¯m→◻¯2bar(Gamma)_(m)rarr bar(â—»)^(2)\bar{\Gamma}_{m} \rightarrow \bar{\square}^{2}Γ¯m→◻¯2 is a cover of ◻¯2◻¯2bar(â—»)^(2)\bar{\square}^{2}◻¯2 in the dividing Nisnevich topology, and becomes an isomorphism after ddddd Nis-localization. In a sense, this allows one to consider the sheaf adNis◻¯adNis◻¯a_(dNis) bar(â—»)a_{\mathrm{dNis}} \bar{\square}adNis◻¯ as a version of a cylinder object and allows many of the constructions of Morel-Voevodsky for a site with interval to go through, although there are occasional technical difficulties that arise.
Definition 5.7. The tensor triangulated category of effective log motives over k,logDMeff(k)k,logDMâ¡eff(k)k,logDM ^(eff)(k)k, \operatorname{logDM}{ }^{\mathrm{eff}}(k)k,logDMâ¡eff(k), is the Verdier localization of the derived category D(PSSTk)DPSSTkD(PSST_(k))D\left(\mathrm{PSST}_{k}\right)D(PSSTk) with respect to the localizing subcategory generated by:
The functor IMeff IMeff IM^("eff ")\mathrm{IM}^{\text {eff }}IMeff shares many of the formal properties of Meff :Smk→DMeff (k)Meff :Smk→DMeff (k)M^("eff "):Sm_(k)rarrDM^("eff ")(k)M^{\text {eff }}: \operatorname{Sm}_{k} \rightarrow \mathrm{DM}^{\text {eff }}(k)Meff :Smk→DMeff (k); we refer the reader to the [19, INTRoDUction] for an overview.
Questions of representing known constructions such as the higher Chow groups with modulus in logDMeff (k)logâ¡DMeff (k)log DM^("eff ")(k)\log \mathrm{DM}^{\text {eff }}(k)logâ¡DMeff (k), or finding direct connections of logDMeff (k)logâ¡DMeff (k)log DM^("eff ")(k)\log \mathrm{DM}^{\text {eff }}(k)logâ¡DMeff (k) with the category MDMeff (k)MDMeff â¡(k)MDM^("eff ")(k)\operatorname{MDM}^{\text {eff }}(k)MDMeff â¡(k) are not discussed in [19]. However, for (X,D)(X,D)(X,D)(X, D)(X,D) a proper modulus pair, one has the log scheme l(X,D)l(X,D)l(X,D)l(X, D)l(X,D), defined using the Deligne-Faltings log structure on XXXXX associated to the ideal sheaf OX(−D)OX(−D)O_(X)(-D)\mathcal{O}_{X}(-D)OX(−D). In general, this is not saturated. Still, there should be presheaves with logloglog\loglog transfers Zltr(X,D)Zltr(X,D)Z_(ltr)(X,D)\mathbb{Z}_{\mathrm{ltr}}(X, D)Zltr(X,D) and Zltrc(X,D)Zltrc(X,D)Z_(ltr)^(c)(X,D)\mathbb{Z}_{\mathrm{ltr}}^{c}(X, D)Zltrc(X,D) using finite and quasi-finite "log correspondences," with value on Y∈1SmkY∈1SmkY in1Sm_(k)Y \in 1 \operatorname{Sm}_{k}Y∈1Smk the free abelian group of integral subschemes WWWWW of Y_×XY_×XY_xx X\underline{Y} \times XY_×X that admit a map of log schemes (WN,(p1∘ν)∗(MY))→l(X,D)WN,p1∘ν∗MY→l(X,D)(W^(N),(p_(1)@nu)^(**)(M_(Y)))rarr l(X,D)\left(W^{N},\left(p_{1} \circ \nu\right)^{*}\left(\mathcal{M}_{Y}\right)\right) \rightarrow l(X, D)(WN,(p1∘ν)∗(MY))→l(X,D), as in the definition of 1Cork(−,−)1Corkâ¡(−,−)1Cor_(k)(-,-)1 \operatorname{Cor}_{k}(-,-)1Corkâ¡(−,−). One could also expect to have presheaves lz(X,D,r)lz(X,D,r)lz(X,D,r)l z(X, D, r)lz(X,D,r) similarly defined, and corresponding to the presheaves with modulus transfers z(X,D,r)z(X,D,r)z(X,D,r)z(X, D, r)z(X,D,r) constructed by Miyazaki. These could be used to give a map
We have briefly mentioned reciprocity sheaves in our discussion of motives with modulus. There is a nice connection of logDMeff (k)logDMâ¡eff (k)logDM ^("eff ")(k)\operatorname{logDM}{ }^{\text {eff }}(k)logDMâ¡eff (k) with the theory of reciprocity sheaves, so we take the opportunity to say a few words about reciprocity sheaves before we describe the theorem of Shuji Saito, which gives the connection between these two theories.
The notion of a reciprocity sheaf and its relation to motives with modulus goes back to the theorem of Rosenlicht-Serre. In our discussion of reciprocity sheaves, we work over a fixed perfect field kkkkk.
Theorem 5.8 (Rosenlicht-Serre [109[109[109[109[109, III]). Let kkkkk be a perfect field, let CCCCC be a smooth complete curve over kkkkk, let GGGGG be an smooth commutative algebraic group over kkkkk, and let fffff : C→GC→GC rarr GC \rightarrow GC→G be a rational map over kkkkk. Let S⊂CS⊂CS sub CS \subset CS⊂C be a finite subset such that fffff is a morphism on C∖SC∖SC\\SC \backslash SC∖S. Then there is an effective divisor DDDDD supported in SSSSS such that, for ggggg a rational function on CCCCC with g≡1modDg≡1modDg-=1mod Dg \equiv 1 \bmod Dg≡1modD, one has
∑P∈C∖SordP(g)⋅f(P)=0∑P∈C∖S ordPâ¡(g)â‹…f(P)=0sum_(P in C\\S)ord_(P)(g)*f(P)=0\sum_{P \in C \backslash S} \operatorname{ord}_{P}(g) \cdot f(P)=0∑P∈C∖SordPâ¡(g)â‹…f(P)=0
in GGGGG.
In [72], reciprocity functors and reciprocity sheaves are defined. We will just give a sketch. One first defines for FFFFF a presheaf with transfers (in the Voevodsky sense), and for a proper modulus pair (X,D)(X,D)(X,D)(X, D)(X,D) with a section a∈F(X∖D)a∈F(X∖D)a in F(X\\D)a \in F(X \backslash D)a∈F(X∖D), what it means for aaaaa to have modulus DDDDD. As an example, if p:C→Xp:C→Xp:C rarr Xp: C \rightarrow Xp:C→X is a non-constant morphism of a smooth proper integral curve CCCCC over kkkkk to XXXXX with p(C)p(C)p(C)p(C)p(C) not contained in DDDDD, and ggggg is a rational function on CCCCC such that g≡1modp∗(D)g≡1modp∗(D)g-=1modp^(**)(D)g \equiv 1 \bmod p^{*}(D)g≡1modp∗(D), then one is required to have
Here, for a 0 -cycle ∑xnx⋅x∑x nxâ‹…xsum_(x)n_(x)*x\sum_{x} n_{x} \cdot x∑xnxâ‹…x on X∖D,a(∑xnx⋅x)=∑xnx⋅px∗ix∗(a)∈F(Speck)X∖D,a∑x nxâ‹…x=∑x nxâ‹…px∗ix∗(a)∈F(Specâ¡k)X\\D,a(sum_(x)n_(x)*x)=sum_(x)n_(x)*p_(x**i_(x)^(**))(a)in F(Spec k)X \backslash D, a\left(\sum_{x} n_{x} \cdot x\right)=\sum_{x} n_{x} \cdot p_{x * i_{x}^{*}}(a) \in F(\operatorname{Spec} k)X∖D,a(∑xnxâ‹…x)=∑xnxâ‹…px∗ix∗(a)∈F(Specâ¡k), where for a closed point xxxxx of X∖D,ix:x→X∖DX∖D,ix:x→X∖DX\\D,i_(x):x rarr X\\DX \backslash D, i_{x}: x \rightarrow X \backslash DX∖D,ix:x→X∖D is the inclusion and px:x→Speckpx:x→Specâ¡kp_(x):x rarr Spec kp_{x}: x \rightarrow \operatorname{Spec} kpx:x→Specâ¡k is the (finite) structure morphism. In general, one imposes a similar condition in F(S)F(S)F(S)F(S)F(S) for a "relative curve" on X×SX×SX xx SX \times SX×S over some smooth base scheme SSSSS.
A presheaf with transfers FFFFF is a reciprocity sheaf if for each quasi-affine UUUUU and section a∈F(U)a∈F(U)a in F(U)a \in F(U)a∈F(U), there is a proper modulus pair (X,D)(X,D)(X,D)(X, D)(X,D) with U=X∖DU=X∖DU=X\\DU=X \backslash DU=X∖D such that aaaaa has modulus DDDDD. Roughly speaking, one should think that each section of FFFFF has "bounded ramification," although the "ramification" for FFFFF itself may be unbounded.
This definition is not quite accurate, as a slightly different notion of "modulus pair" from what we have defined here is used in [72]. A more elegant definition of reciprocity sheaf is given in [73]. This new notion is a bit more restrictive than the old one, but by [73, THEOREM 2], the two notions agree on for F∈MNSTk_F∈MNSTk_F inMNST_(k)_F \in \underline{\mathrm{MNST}_{k}}F∈MNSTk_.
Using the definition of [73], the reciprocity sheaves define a the full subcategory RSTkRSTkRST_(k)\mathbf{R S T}_{k}RSTk of PSTkPSTkPST_(k)\mathrm{PST}_{k}PSTk, strictly containing the subcategory HIk⊂PSTkHIk⊂PSTkHI_(k)subPST_(k)\mathbf{H I}_{k} \subset \mathrm{PST}_{k}HIk⊂PSTk of A1A1A^(1)\mathbb{A}^{1}A1-homotopy invariant presheaves with transfer. There is also the subcategory RSTNis,kRSTNis,kRST_(Nis,k)\mathbf{R S T}_{\mathrm{Nis}, k}RSTNis,k of NSTkNSTkNST_(k)\mathrm{NST}_{k}NSTk, consisting of those reciprocity presheaves that are Nisnevich sheaves.
Here is the promised theorem of Saito. For a sheaf G∈NNSTkG∈NNSTkG inNNST_(k)G \in \operatorname{NNST}_{k}G∈NNSTk, we say that GGGGG is
HdNis∗(X,G∣XdNis)→HdNis∗(X×◻¯,G∣X×◻¯dNis)HdNis∗X,G∣XdNis→HdNis∗X×◻¯,G∣X×◻¯dNisH_(dNis)^(**)(X,G_(∣X_(dNis)))rarrH_(dNis)^(**)(X xx( bar(â—»)),G_(∣X xx bar(â—»)_(dNis)))H_{d \mathrm{Nis}}^{*}\left(X, G_{\mid X_{d \mathrm{Nis}}}\right) \rightarrow H_{d \mathrm{Nis}}^{*}\left(X \times \bar{\square}, G_{\mid X \times \bar{\square}_{d \mathrm{Nis}}}\right)HdNis∗(X,G∣XdNis)→HdNis∗(X×◻¯,G∣X×◻¯dNis)
induced by the projection X×◻¯→XX×◻¯→XX xx bar(â—»)rarr XX \times \bar{\square} \rightarrow XX×◻¯→X is an isomorphism. Here ddddd Nis refers to the divided Nisnevich site.
Theorem 5.9 (Saito [105, THEOREM 0.2]). There exists a fully faithful exact functor
log:RSTNis,k→lNSTklog:RSTNis,k→lNSTklog:RST_(Nis,k)rarrlNST_(k)\log : \mathbf{R S T}_{\mathrm{Nis}, k} \rightarrow \operatorname{lNST}_{k}log:RSTNis,k→lNSTk
such that log(F)logâ¡(F)log(F)\log (F)logâ¡(F) is strictly ◻¯â—»Â¯bar(â—»)\bar{\square}◻¯-invariantfor every F∈RSTNis,kF∈RSTNis,kF inRST_(Nis,k)F \in \mathbf{R S T}_{\mathrm{Nis}, k}F∈RSTNis,k. Moreover, for each X∈SmkX∈SmkX inSm_(k)X \in \operatorname{Sm}_{k}X∈Smk, there is a natural isomorphism
On the other hand, if kkkkk has characteristic p>0p>0p > 0p>0p>0, we have the isomorphism [52] of the Nisnevich sheaves on SmkSmkSm_(k)\mathrm{Sm}_{k}Smk,
Our goal in this section is to give some details of the story sketched above.
We first discuss the papers of Bloch-Kato, Fontaine-Messing, Kurihara and Sato without reference to all the advances in ppppp-adic Hodge theory that followed these works; we wanted to give the reader just enough background to put the connections with motivic
cohomology in context. We will then describe the works of Geisser and Zhong, as well as results of Geisser-Hesselholt that form some of the foundations for the work of BhattMorrow-Scholze. We conclude with a description of the Bhatt-Morrow-Scholze motivic tower and its connection with the ppppp-adic cycle complexes.
We refer the reader to [15] for background on crystalline cohomology.
6.1. A quick overview of some ppp\boldsymbol{p}p-adic Hodge theory
We begin with a few comments on the paper of Bloch and Kato [28], which we have already mentioned in our discussion of the Beilinson-Lichtenbaum conjectures. They consider the spectrum SSSSS of a complete dvrΛdvrâ¡Î›dvr Lambda\operatorname{dvr} \Lambdadvrâ¡Î› with generic point η=SpecK↪Sη=Specâ¡K↪Seta=Spec K↪S\eta=\operatorname{Spec} K \hookrightarrow Sη=Specâ¡K↪S and closed point s=Speck↪Ss=Specâ¡k↪Ss=Spec k↪Ss=\operatorname{Spec} k \hookrightarrow Ss=Specâ¡k↪S, and a smooth and proper SSSSS-scheme X→SX→SX rarr SX \rightarrow SX→S with generic fiber V:=XηV:=XηV:=X_(eta)V:=X_{\eta}V:=Xη and special fiber Y:=Xs.V¯,Y¯Y:=Xs.V¯,Y¯Y:=X_(s). bar(V), bar(Y)Y:=X_{s} . \bar{V}, \bar{Y}Y:=Xs.V¯,Y¯ denote V,YV,YV,YV, YV,Y over the respective algebraic closures K¯K¯bar(K)\bar{K}K¯ and k¯k¯bar(k)\bar{k}k¯ of KKKKK and kkkkk. Let Λ¯Î›Â¯bar(Lambda)\bar{\Lambda}Λ¯ be the integral closure of ΛΛLambda\LambdaΛ in K¯,S¯:=SpecΛ¯K¯,S¯:=Specâ¡Î›Â¯bar(K), bar(S):=Spec bar(Lambda)\bar{K}, \bar{S}:=\operatorname{Spec} \bar{\Lambda}K¯,S¯:=Specâ¡Î›Â¯, and X¯=X×SS¯X¯=X×SS¯bar(X)=Xxx_(S) bar(S)\bar{X}=X \times_{S} \bar{S}X¯=X×SS¯. Let G=Gal(K¯/K)G=Galâ¡(K¯/K)G=Gal( bar(K)//K)G=\operatorname{Gal}(\bar{K} / K)G=Galâ¡(K¯/K) and let CCCCC denote the completion of K¯K¯bar(K)\bar{K}K¯.
The closure Y¯Y¯bar(Y)\bar{Y}Y¯ has its crystalline cohomology Hcrys ∗(Y¯/W(k¯))Hcrys ∗(Y¯/W(k¯))H_("crys ")^(**)( bar(Y)//W( bar(k)))H_{\text {crys }}^{*}(\bar{Y} / W(\bar{k}))Hcrys ∗(Y¯/W(k¯)) with action of Frobenius, giving the pipip^(i)p^{i}pi-eigenspace
We have the inclusions i¯:Y¯→X¯,j¯:V¯→X¯i¯:Y¯→X¯,j¯:V¯→X¯bar(i): bar(Y)rarr bar(X), bar(j): bar(V)rarr bar(X)\bar{i}: \bar{Y} \rightarrow \bar{X}, \bar{j}: \bar{V} \rightarrow \bar{X}i¯:Y¯→X¯,j¯:V¯→X¯ and the spectral sequence
inducing a descending filtration F∗Het∗(V¯,Qp)F∗Het∗V¯,QpF^(**)H_(et)^(**)(( bar(V)),Q_(p))F^{*} H_{\mathrm{et}}^{*}\left(\bar{V}, \mathbb{Q}_{p}\right)F∗Het∗(V¯,Qp) on Het∗(V¯,Qp)Het∗V¯,QpH_(et)^(**)(( bar(V)),Q_(p))H_{\mathrm{et}}^{*}\left(\bar{V}, \mathbb{Q}_{p}\right)Het∗(V¯,Qp) with F0Hq=HqF0Hq=HqF^(0)H^(q)=H^(q)F^{0} H^{q}=H^{q}F0Hq=Hq and Fq+1Hq=0Fq+1Hq=0F^(q+1)H^(q)=0F^{q+1} H^{q}=0Fq+1Hq=0.
Theorem 6.1 (Bloch-Kato [28, тHEORem 0.7]). Suppose that kkkkk is perfect and that YYYYY is ordinary. Then there are natural GGGGG-equivariant isomorphisms
The second result is a special case of the Bloch-Kato conjecture.
Theorem 6.3 (Bloch-Kato [28, THEOREM 5.12]). Let F be a henselian discretely valued field of characteristic 0 , with residue field of characteristic p>0p>0p > 0p>0p>0. Then the Galois symbol
To set this up, they consider the syntomic topology on SchWn(k)SchWn(k)Sch_(W_(n)(k))\mathrm{Sch}_{W_{n}(k)}SchWn(k), where a cover is a surjective syntomic map (we described syntomic maps in Section 3.3). The crystalline structure sheaf Oncrys Oncrys O_(n)^("crys ")\mathcal{O}_{n}^{\text {crys }}Oncrys defines a sheaf for the syntomic topology with a surjection to the usual structure sheaf OnOnO_(n)\mathcal{O}_{n}On on SchWn(k)SchWn(k)Sch_(W_(n)(k))\operatorname{Sch}_{W_{n}(k)}SchWn(k). Letting JnJnJ_(n)J_{n}Jn denote the kernel of Oncrys →OnOncrys →OnO_(n)^("crys ")rarrO_(n)\mathcal{O}_{n}^{\text {crys }} \rightarrow \mathcal{O}_{n}Oncrys →On, one has the rrrrr th divided power Jn[r]Jn[r]J_(n)^([r])J_{n}^{[r]}Jn[r]; this gives us the sheaf S~nrS~nrtilde(S)_(n)^(r)\tilde{S}_{n}^{r}S~nr defined as the kernel of ϕ−pr:Jn[r]→Oncrys ϕ−pr:Jn[r]→Oncrys phi-p^(r):J_(n)^([r])rarrO_(n)^("crys ")\phi-p^{r}: J_{n}^{[r]} \rightarrow \mathcal{O}_{n}^{\text {crys }}ϕ−pr:Jn[r]→Oncrys . Modifying this by taking the image SnrSnrS_(n)^(r)S_{n}^{r}Snr of the reduction map S~n+rr→S~nrS~n+rr→S~nrtilde(S)_(n+r)^(r)rarr tilde(S)_(n)^(r)\tilde{S}_{n+r}^{r} \rightarrow \tilde{S}_{n}^{r}S~n+rr→S~nr gives the inverse system {Snr}nSnrn{S_(n)^(r)}_(n)\left\{S_{n}^{r}\right\}_{n}{Snr}n and the cohomology
The ring Bcrys +Bcrys +B_("crys ")^(+)B_{\text {crys }}^{+}Bcrys +is defined as follows. The characteristic ppppp ring OK¯/pOK¯/pO_( bar(K))//p\mathcal{O}_{\bar{K}} / pOK¯/p forms an inverse system via the Frobenius endomorphism; let
a perfect characteristic ppppp ring. We have the ring of truncated Witt vectors Wn(Ob)WnObW_(n)(O^(b))W_{n}\left(\mathcal{O}^{\mathrm{b}}\right)Wn(Ob) and a surjection πn:Wn(Ob)→OK¯/pnÏ€n:WnOb→OK¯/pnpi_(n):W_(n)(O^(b))rarrO_( bar(K))//p^(n)\pi_{n}: W_{n}\left(\mathcal{O}^{b}\right) \rightarrow \mathcal{O}_{\bar{K}} / p^{n}Ï€n:Wn(Ob)→OK¯/pn. Let WnDP(OK¯)WnDPOK¯W_(n)^(DP)(O_( bar(K)))W_{n}^{\mathrm{DP}}\left(\mathcal{O}_{\bar{K}}\right)WnDP(OK¯) be the divided power envelope of the kernel of πnÏ€npi_(n)\pi_{n}Ï€n, forming the inverse system {WnDP(OK¯)}n≥0WnDPOK¯n≥0{W_(n)^(DP)(O_( bar(K)))}_(n >= 0)\left\{W_{n}^{\mathrm{DP}}\left(\mathcal{O}_{\bar{K}}\right)\right\}_{n \geq 0}{WnDP(OK¯)}n≥0. Let
The Frobenius on Wn(Ob)WnObW_(n)(O^(b))W_{n}\left(\mathcal{O}^{\mathrm{b}}\right)Wn(Ob) induces a Frobenius on Bcrys +Bcrys +B_("crys ")^(+)B_{\text {crys }}^{+}Bcrys +and the filtration Jn[∗]Jn[∗]J_(n)^([**])J_{n}^{[*]}Jn[∗] of WnDP(OK¯)WnDPOK¯W_(n)^(DP)(O_( bar(K)))W_{n}^{\mathrm{DP}}\left(\mathcal{O}_{\bar{K}}\right)WnDP(OK¯) induces a filtration Fil* Bcrys +Bcrys +B_("crys ")^(+)B_{\text {crys }}^{+}Bcrys +on Bcrys +Bcrys +B_("crys ")^(+)B_{\text {crys }}^{+}Bcrys +.
The derived push-forward of the complex Jn[r]→ϕ−prOncrys Jn[r]→ϕ−prOncrys J_(n)^([r])rarr"phi-p^(r)"O_(n)^("crys ")J_{n}^{[r]} \xrightarrow{\phi-p^{r}} \mathcal{O}_{n}^{\text {crys }}Jn[r]→ϕ−prOncrys is an analog of the Deligne complex, as expressed in the following theorem.
Theorem 6.4 ([40, corollary to theorem 1.6, LemMA 3.1]). Suppose that XXXXX is admissible 33^(3){ }^{3}3 and Λ=W(k)Λ=W(k)Lambda=W(k)\Lambda=W(k)Λ=W(k). Then for m≤r<pm≤r<pm <= r < pm \leq r<pm≤r<p there is an exact sequence
by sending a pnpnp^(n)p^{n}pn-root of unity εεepsi\varepsilonε in K¯K¯bar(K)\bar{K}K¯ to the logarithm of the Teichmüller lift of the mod ppppp reduction of εεepsi\varepsilonε, and passing to the limit in nnnnn. Let t∈Qp(1)(K¯)t∈Qp(1)(K¯)t inQ_(p)(1)( bar(K))t \in \mathbb{Q}_{p}(1)(\bar{K})t∈Qp(1)(K¯) be a nonzero element and define Bcrys =Bcrys +[1/t]Bcrys =Bcrys +[1/t]B_("crys ")=B_("crys ")^(+)[1//t]B_{\text {crys }}=B_{\text {crys }}^{+}[1 / t]Bcrys =Bcrys +[1/t], with induced filtration and Galois action. Twisting with respect to ttttt translates the twisted version to the untwisted one.
Theorem 6.5 (Kurihara). Suppose that [k:kp]<∞k:kp<∞[k:k^(p)] < oo\left[k: k^{p}\right]<\infty[k:kp]<∞. Let X→SX→SX rarr SX \rightarrow SX→S be smooth and projective and suppose that r<p−1r<p−1r < p-1r<p-1r<p−1. Then there is a distinguished triangle in D(Yet^)DYet^D(Y_( hat(et)))D\left(Y_{\hat{e t}}\right)D(Yet^),
which recovers the one in Kurihara's theorem for r<p−1r<p−1r < p-1r<p-1r<p−1 by applying i∗i∗i^(**)i^{*}i∗.
Using a similar method, Schneider's construction was extended to the semi-stable case by Sato [106], who defines the object Tn(r)∈D(Xet )Tn(r)∈DXet T_(n)(r)in D(X_("et "))\mathfrak{T}_{n}(r) \in D\left(X_{\text {et }}\right)Tn(r)∈D(Xet ) with Tn(r)≅Sn(r)Tn(r)≅Sn(r)T_(n)(r)~=S_(n)(r)\mathfrak{T}_{n}(r) \cong S_{n}(r)Tn(r)≅Sn(r) in the smooth case.
6.2. Étale motivic cohomology
We return to algebraic cycles. As before, we consider a smooth separated finite type SSSSS-scheme X→S=SpecΛX→S=Specâ¡Î›X rarr S=Spec LambdaX \rightarrow S=\operatorname{Spec} \LambdaX→S=Specâ¡Î› with generic fiber j:V→Xj:V→Xj:V rarr Xj: V \rightarrow Xj:V→X and special fiber i:Y→Xi:Y→Xi:Y rarr Xi: Y \rightarrow Xi:Y→X, and with ΛΛLambda\LambdaΛ a mixed characteristic (0,p)(0,p)(0,p)(0, p)(0,p) dvr with perfect residue field.
Geisser [49] considers the motivic complex Z(r)XZ(r)XZ(r)_(X)\mathbb{Z}(r)_{X}Z(r)X on a smooth SSSSS-scheme X→SX→SX rarr SX \rightarrow SX→S as a sheaf of complexes on XNisXNisX_(Nis)X_{\mathrm{Nis}}XNis. Here we use the reindexed Bloch cycle complex to define Z(r)X∗(U)Z(r)X∗(U)Z(r)_(X)^(**)(U)\mathbb{Z}(r)_{X}^{*}(U)Z(r)X∗(U) as
Theorem 6.6 (Geisser [49, THEOREM 1.3]). Let X→X→X rarrX \rightarrowX→ Spec ΛΛLambda\LambdaΛ be smooth and essentially of finite type, with ΛΛLambda\LambdaΛ a complete discrete valuation ring of mixed characteristic (0,p)(0,p)(0,p)(0, p)(0,p). Then there is a distinguished triangle in Db(Xèt )DbXèt D^(b)(X_("èt "))D^{b}\left(X_{\text {èt }}\right)èDb(Xèt ),
and an isomorphism Z/pn(r)ét≅Sn(r)Z/pn(r)eÌt≅Sn(r)Z//p^(n)(r)_(eÌt)~=S_(n)(r)\mathbb{Z} / p^{n}(r)_{e Ì t} \cong S_{n}(r)Z/pn(r)eÌt≅Sn(r) in Db(Xe˙t)DbXeË™tD^(b)(X_(e^(Ë™)t))D^{b}\left(X_{\dot{e} t}\right)Db(XeË™t) that transforms this triangle to Schneider's defining triangle (6.3).
Zhong has extended this to the semi-stable case, establishing an isomorphism with Sato's construction In(r)In(r)I_(n)(r)\mathfrak{I}_{n}(r)In(r) after a truncation [128, PROPOSITION 4.5]:
The trace map trc : Ki(Z/p⋅)→TYiKiZ/p⋅→TYiK_(i)(Z//p^(*))rarrTY_(i)\mathcal{K}_{i}\left(\mathbb{Z} / p^{\cdot}\right) \rightarrow \mathcal{T} \mathcal{Y}_{i}Ki(Z/pâ‹…)→TYi is an isomorphism of pro-sheaves on SmkétSmkeÌtSm_(keÌt)\mathrm{Sm}_{k e Ì t}SmkeÌt.
Now consider a smooth finite type scheme X→X→X rarrX \rightarrowX→ Spec ΛΛLambda\LambdaΛ with special fiber i:Y→Xi:Y→Xi:Y rarr Xi: Y \rightarrow Xi:Y→X and generic fiber j:V→Xj:V→Xj:V rarr Xj: V \rightarrow Xj:V→X, as before.
Theorem 6.8 (Geisser-Hesselholt [51, THEorems A and B]). Suppose ΛΛLambda\LambdaΛ is henselian.
A. Suppose X→SpecΛX→Specâ¡Î›X rarr Spec LambdaX \rightarrow \operatorname{Spec} \LambdaX→Specâ¡Î› is smooth and proper. Then
is an isomorphism for all q∈Zq∈Zq inZq \in \mathbb{Z}q∈Z and v≥1v≥1v >= 1v \geq 1v≥1.
B. Suppose that X→SpecΛX→Specâ¡Î›X rarr Spec LambdaX \rightarrow \operatorname{Spec} \LambdaX→Specâ¡Î› is smooth and finite type. Then the map of prosheaves on YétYeÌtY_(eÌt)Y_{e Ì t}YeÌt,
is an isomorphism for all q∈Zq∈Zq inZq \in \mathbb{Z}q∈Z and all ν≥1ν≥1nu >= 1\nu \geq 1ν≥1.
Remark 6.9. To pass from the isomorphism of Theorem 6.7 to that of Theorem 6.8(B), Geisser-Hesselholt rely on the theorem of McCarthy [88], stating that the cyclotomic trace map from relative KKKKK-theory to relative TC,
trc:Kq(X/πn,X/πn−r,Z/pν)→TCq(X/πn,X/πn−r,Z/pν)trc:KqX/Ï€n,X/Ï€n−r,Z/pν→TCqX/Ï€n,X/Ï€n−r,Z/pνtrc:K_(q)(X//pi^(n),X//pi^(n-r),Z//p^(nu))rarrTC_(q)(X//pi^(n),X//pi^(n-r),Z//p^(nu))\operatorname{trc}: K_{q}\left(X / \pi^{n}, X / \pi^{n-r}, \mathbb{Z} / p^{\nu}\right) \rightarrow \mathrm{TC}_{q}\left(X / \pi^{n}, X / \pi^{n-r}, \mathbb{Z} / p^{\nu}\right)trc:Kq(X/Ï€n,X/Ï€n−r,Z/pν)→TCq(X/Ï€n,X/Ï€n−r,Z/pν)
is an isomorphism for affine XXXXX. Thus, the KKKKK-theory and topological cyclic homology of non-reduced schemes play a central role in the proof of Theorem 6.8.
6.4. Integral ppppp-adic Hodge theory and the motivic filtration
Let CpCpC_(p)\mathbb{C}_{p}Cp be the completion of the algebraic closure of QpQpQ_(p)\mathbb{Q}_{p}Qp, with ring of integers OCpOCpO_(C_(p))\mathcal{O}_{\mathbb{C}_{p}}OCp. As in our review of the work of Fontaine-Messing, we have the FpFpF_(p)\mathbb{F}_{p}Fp-algebra OCp/pOCp/pO_(C_(p))//p\mathcal{O}_{\mathbb{C}_{p}} / pOCp/p, its perfection OCpbOCpbO_(C_(p))^(b)\mathcal{O}_{\mathbb{C}_{p}}^{b}OCpb and the ring of Witt vectors Ainf (OCp):=W(OCpb)Ainf OCp:=WOCpbA_("inf ")(O_(C_(p))):=W(O_(C_(p))^(b))A_{\text {inf }}\left(\mathcal{O}_{\mathbb{C}_{p}}\right):=W\left(\mathcal{O}_{\mathbb{C}_{p}}^{b}\right)Ainf (OCp):=W(OCpb). Hesselholt has connected this with negative cyclic homology TC−TC−TC^(-)\mathrm{TC}^{-}TC−, constructing an isomorphism
This has been generalized by Bhatt-Morrow-Scholze in the setting of perfectoid rings (see [17, DEFINITION 3.5]). For a perfectoid ring RRRRR, we have Scholze's ring RbRbR^(b)R^{b}Rb, defined as for OCpbOCpbO_(C_(p))^(b)\mathcal{O}_{\mathbb{C}_{p}}^{b}OCpb by taking the perfection of R/pR/pR//pR / pR/p,
This gives the ring of Witt vectors Ainf (R):=W(Rb)Ainf (R):=WRbA_("inf ")(R):=W(R^(b))A_{\text {inf }}(R):=W\left(R^{b}\right)Ainf (R):=W(Rb) with Frobenius ϕÏ•phi\phiÏ• induced by the Frobenius on RbRbR^(b)R^{b}Rb.
Theorem 6.10 (Bhatt-Morrow-Scholze [18, THEOREM 1.6]). Let RRRRR be a perfectoid ring. Then there is a canonical ϕÏ•phi\phiÏ•-equivariant isomorphism π0TC−(R,Zp)≅Ainf (R)Ï€0TC−R,Zp≅Ainf (R)pi_(0)TC^(-)(R,Z_(p))~=A_("inf ")(R)\pi_{0} \mathrm{TC}^{-}\left(R, \mathbb{Z}_{p}\right) \cong A_{\text {inf }}(R)Ï€0TC−(R,Zp)≅Ainf (R).
Fix a discretely valued extension KKKKK of QpQpQ_(p)\mathbb{Q}_{p}Qp, with ring of integers OKOKO_(K)\mathcal{O}_{K}OK having perfect residue field kkkkk. Let CCCCC be the completed algebraic closure of KKKKK, with ring of integers OCOCO_(C)\mathcal{O}_{C}OC. Let Ainf :=Ainf (OC)Ainf :=Ainf OCA_("inf "):=A_("inf ")(O_(C))A_{\text {inf }}:=A_{\text {inf }}\left(\mathcal{O}_{C}\right)Ainf :=Ainf (OC).
Theorem 6.11 ([18, THEOREM 1.8]). Let A~A~tilde(A)\tilde{A}A~ be an OCOCO_(C)\mathcal{O}_{C}OC-algebra that can be written as the ppppp-adic completion of a smooth OCOCO_(C)\mathcal{O}_{C}OC-algebra. There is a functorial (in A~A~tilde(A)\tilde{A}A~ ) ϕÏ•phi\phiÏ•-equivariant isomorphism of E∞−Ainf E∞−Ainf E_(oo)-A_("inf ")E_{\infty}-A_{\text {inf }}E∞−Ainf -algebras
The Postnikov tower τ≥∗TC(−;Zp)τ≥∗TC−;Zptau_( >= **)TC(-;Z_(p))\tau_{\geq *} \mathrm{TC}\left(-; \mathbb{Z}_{p}\right)τ≥∗TC(−;Zp) for the presheaf of spectra TC(−;Zp)TC−;ZpTC(-;Z_(p))\mathrm{TC}\left(-; \mathbb{Z}_{p}\right)TC(−;Zp) on A~qsyn A~qsyn tilde(A)_("qsyn ")\tilde{A}_{\text {qsyn }}A~qsyn induces the tower over TC(A~;ZpTCâ¡A~;ZpTC(( tilde(A));Z_(p):}\operatorname{TC}\left(\tilde{A} ; \mathbb{Z}_{p}\right.TCâ¡(A~;Zp :
(see [18,$1.4])[18,$1.4])[18,$1.4])[18, \$ 1.4])[18,$1.4]). Define the sheaves ZpBMS(n)ZpBMS(n)Z_(p)^(BMS)(n)\mathbb{Z}_{p}^{\mathrm{BMS}}(n)ZpBMS(n) by sheafifying the presheaf
ZpBMS(r)X≅τ≤rRψZp(r)X,étZpBMS(r)X≅τ≤rRψZp(r)X,eÌtZ_(p)^(BMS)(r)X~=tau_( <= r)R psiZ_(p)(r)_(X,eÌt)\mathbb{Z}_{p}^{\mathrm{BMS}}(r) X \cong \tau_{\leq r} R \psi \mathbb{Z}_{p}(r)_{X, e Ì t}ZpBMS(r)X≅τ≤rRψZp(r)X,eÌt